(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. The line below identifies what version of Mathematica created this file, but it can be opened using any other version as well."; FrontEndVersion = "X Window System Mathematica Notebook Front End Version 2.2"; X11StandardFontEncoding; fontset = title, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e8, 24, fontName, "times"; fontset = subtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e6, 18, fontName, "times"; fontset = subsubtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, e6, 14, fontName, "times"; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, a20, 18, fontName, "times"; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, a15, 14, fontName, "times"; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, a12, 12, fontName, "times"; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, fontName, "times"; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, fontName, "times"; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, 12, fontName, "courier"; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, fontName, "courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, 12, fontName, "courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, 12, fontName, "courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, 12, fontName, "courier"; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, 12, fontName, "courier"; fontset = name, inactive, noPageBreakInGroup, nohscroll, preserveAspect, M7, italic, B65535, 10, fontName, "times"; fontset = header, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, 12, fontName, "times"; fontset = leftheader, 12, fontName, "times"; fontset = footer, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, italic, 12, fontName, "times"; fontset = leftfooter, 12, fontName, "times"; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, fontName, "times"; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, fontName, "times"; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, fontName, "courier"; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, fontName, "times"; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, fontName, "times"; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, fontName, "times"; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, fontName, "times"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, fontName, "times";paletteColors = 128; automaticGrouping; currentKernel; ] :[font = title; inactive; preserveAspect; startGroup] Single equations without initial conditions :[font = title; inactive; preserveAspect] First order equations ;[s] 1:0,0;21,-1; 1:1,0,0 ,times,1,18,0,0,0; :[font = section; inactive; preserveAspect; startGroup] Equation (1) :[font = input; preserveAspect; startGroup] ode = (x^4-x^3)*u'[x]+2*x^4*u[x] == x^3/3+C :[font = output; output; inactive; preserveAspect; endGroup] 2*x^4*u[x] + (-x^3 + x^4)*Derivative[1][u][x] == C + x^3/3 ;[o] 3 4 3 4 x 2 x u[x] + (-x + x ) u'[x] == C + -- 3 :[font = input; preserveAspect; startGroup] DSolve[ ode,u,x ] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] {{u -> Function[x, (6*C - 3*x^2 + 2*x^3)/ (12*(-1 + x)^2*x^2) + E^(-2*x - 2*Log[1 - x])*C[1]]}} ;[o] 2 3 6 C - 3 x + 2 x {{u -> Function[x, ----------------- + 2 2 12 (-1 + x) x -2 x - 2 Log[1 - x] E C[1]]}} :[font = section; inactive; pageBreak; preserveAspect; startGroup] Equation (2) :[font = input; preserveAspect; startGroup] ode = -1/2 u'[x] + u[x] == Sin[x] :[font = output; output; inactive; preserveAspect; endGroup] u[x] - Derivative[1][u][x]/2 == Sin[x] ;[o] u'[x] u[x] - ----- == Sin[x] 2 :[font = input; preserveAspect; startGroup] DSolve[ ode,u,x ] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] {{u -> Function[x, E^(2*x)*C[1] + (2*(Cos[x] + 2*Sin[x]))/5]}} ;[o] 2 x 2 (Cos[x] + 2 Sin[x]) {{u -> Function[x, E C[1] + ---------------------]}} 5 :[font = section; inactive; preserveAspect; startGroup] Equation (3) :[font = input; preserveAspect; startGroup] ode = y'[x] == y[x]/(y[x] Log[y[x]]+x) :[font = output; output; inactive; preserveAspect; endGroup] Derivative[1][y][x] == y[x]/(x + Log[y[x]]*y[x]) ;[o] y[x] y'[x] == ------------------ x + Log[y[x]] y[x] :[font = input; preserveAspect; startGroup] DSolve[ ode,y,x ] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] DSolve[Derivative[1][y][x] == y[x]/(x + Log[y[x]]*y[x]), y, x] ;[o] y[x] DSolve[y'[x] == ------------------, y, x] x + Log[y[x]] y[x] :[font = section; inactive; preserveAspect; startGroup] Equation (4) :[font = input; preserveAspect; startGroup] ode = 2 y[x] y'[x]^2 - 2 x y'[x] - y[x] == 0 :[font = output; output; inactive; preserveAspect; endGroup] -y[x] - 2*x*Derivative[1][y][x] + 2*y[x]*Derivative[1][y][x]^2 == 0 ;[o] 2 -y[x] - 2 x y'[x] + 2 y[x] y'[x] == 0 :[font = input; preserveAspect; startGroup] DSolve[ ode,y,x ] :[font = message; inactive; preserveAspect] DSolve::dnim: Built-in procedures cannot solve this differential equation. :[font = output; output; inactive; preserveAspect; endGroup; endGroup] DSolve[-y[x] - 2*x*Derivative[1][y][x] + 2*y[x]*Derivative[1][y][x]^2 == 0, y, x] ;[o] 2 DSolve[-y[x] - 2 x y'[x] + 2 y[x] y'[x] == 0, y, x] :[font = section; inactive; preserveAspect; startGroup] Equation (5) :[font = input; preserveAspect; startGroup] ode = y'[x] + y[x] == y[x]^3 Sin[x] :[font = output; output; inactive; preserveAspect; endGroup] y[x] + Derivative[1][y][x] == Sin[x]*y[x]^3 ;[o] 3 y[x] + y'[x] == Sin[x] y[x] :[font = input; preserveAspect; startGroup] DSolve[ ode,y,x ] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] {{y -> Function[x, -(E^(2*x)*C[1] + (2*Cos[x])/5 + (4*Sin[x])/5)^(-1/2)]}, {y -> Function[x, (E^(2*x)*C[1] + (2*Cos[x])/5 + (4*Sin[x])/5)^(-1/2)]}, {y -> Function[x, 0]}} ;[o] {{y -> Function[x, -( 1 -------------------------------------)]}, 2 x 2 Cos[x] 4 Sin[x] Sqrt[E C[1] + -------- + --------] 5 5 {y -> Function[x, 1 -------------------------------------]}, 2 x 2 Cos[x] 4 Sin[x] Sqrt[E C[1] + -------- + --------] 5 5 {y -> Function[x, 0]}} :[font = section; inactive; preserveAspect; startGroup] Equation (6) :[font = input; preserveAspect; startGroup] ode = y'[x] + P[x] y[x] == Q[x] y[x]^n :[font = output; output; inactive; preserveAspect; endGroup] P[x]*y[x] + Derivative[1][y][x] == Q[x]*y[x]^n ;[o] n P[x] y[x] + y'[x] == Q[x] y[x] :[font = text; inactive; preserveAspect; plain; bold] The general form of the Bernoulli equation: :[font = input; preserveAspect; startGroup] DSolve[ ode,y,x ] :[font = output; output; inactive; preserveAspect; endGroup] {{y -> Function[x, (E^((-1 + n)*Integrate[P[x], x])* (C[1] + (1 - n)*Integrate[E^ ((1 - n)*Integrate[P[x], x])*Q[x], x]))^ (1 - n)^(-1)]}} ;[o] (-1 + n) Integrate[P[x], x] {{y -> Function[x, Power[E (C[1] + (1 - n) Integrate[E (1 - n) Integrate[P[x], x] Q[x], x]), 1/(1 - n)]]}} :[font = text; inactive; preserveAspect; plain; bold] Special case: n = 0 :[font = input; preserveAspect; startGroup] DSolve[ ode /. n->0, y,x ] :[font = message; inactive; preserveAspect] General::intinit: Loading integration packages -- please wait. :[font = output; output; inactive; preserveAspect; endGroup] {{y -> Function[x, C[1]/E^Integrate[P[x], x] + Integrate[E^Integrate[P[DSolve`t], DSolve`t]* Q[DSolve`t], {DSolve`t, 0, x}]/E^Integrate[P[x], x] ]}} ;[o] C[1] {{y -> Function[x, ------------------- + Integrate[P[x], x] E Integrate[P[DSolve`t], DSolve`t] Integrate[E Q[DSolve`t], {DSolve`t, 0, x}] / Integrate[P[x], x] E ]}} :[font = text; inactive; preserveAspect; plain; bold] Special case: n = 1 :[font = input; preserveAspect; startGroup] DSolve[ ode /. n->1, y,x ] :[font = output; output; inactive; preserveAspect; endGroup] {{y -> Function[x, E^Integrate[-P[x] + Q[x], x]*C[1]]}} ;[o] Integrate[-P[x] + Q[x], x] {{y -> Function[x, E C[1]]}} :[font = input; preserveAspect; startGroup] DSolve[ ode /. n->Pi, y,x] :[font = output; output; inactive; preserveAspect; endGroup] {{y -> Function[x, (E^((-1 + Pi)*Integrate[P[x], x])* (C[1] + (1 - Pi)* Integrate[E^((1 - Pi)*Integrate[P[x], x])*Q[x], x]))^(1 - Pi)^(-1)]}} ;[o] (-1 + Pi) Integrate[P[x], x] {{y -> Function[x, Power[E (C[1] + (1 - Pi) Integrate[E (1 - Pi) Integrate[P[x], x] Q[x], x]), 1/(1 - Pi)]]}} :[font = text; inactive; preserveAspect] And handles n to be complex valued too: ;[s] 1:0,0;39,-1; 1:1,0,0 ,times,1,12,0,0,0; :[font = input; preserveAspect; startGroup] DSolve[ ode /. n->I, y,x ] :[font = message; inactive; preserveAspect] Solve::ifun: Warning: Inverse functions are being used by Solve, so some solutions may not be found. :[font = output; output; inactive; preserveAspect; endGroup; endGroup] {{y -> Function[x, (E^((-1 + I)*Integrate[P[x], x])* (C[1] + (1 - I)*Integrate[E^ ((1 - I)*Integrate[P[x], x])*Q[x], x]))^ (1/2 + I/2)]}} ;[o] (-1 + I) Integrate[P[x], x] {{y -> Function[x, Power[E (C[1] + (1 - I) Integrate[E (1 - I) Integrate[P[x], x] Q[x], x]), 1/2 + I/2]]}} :[font = section; inactive; preserveAspect; startGroup] Equation (7) :[font = input; preserveAspect; startGroup] ode= (x^2-1) y'[x]^2 - 2 x y[x]y'[x] + y^2 - 1 == 0 :[font = output; output; inactive; preserveAspect; endGroup] -1 + y^2 - 2*x*y[x]*Derivative[1][y][x] + (-1 + x^2)*Derivative[1][y][x]^2 == 0 ;[o] 2 2 2 -1 + y - 2 x y[x] y'[x] + (-1 + x ) y'[x] == 0 :[font = input; preserveAspect; startGroup] DSolve[ ode,y,x ] :[font = message; inactive; preserveAspect] DSolve::dnim: Built-in procedures cannot solve this differential equation. :[font = output; output; inactive; preserveAspect; endGroup; endGroup] DSolve[-1 + y^2 - 2*x*y[x]*Derivative[1][y][x] + (-1 + x^2)*Derivative[1][y][x]^2 == 0, y, x] ;[o] 2 2 2 DSolve[-1 + y - 2 x y[x] y'[x] + (-1 + x ) y'[x] == 0, y, x] :[font = section; inactive; preserveAspect; startGroup] Equation (8) :[font = input; preserveAspect; startGroup] ode = f[x y'[x]] == g[y'[x]] :[font = output; output; inactive; preserveAspect; endGroup] f[x*Derivative[1][y][x]] == g[Derivative[1][y][x]] ;[o] f[x y'[x]] == g[y'[x]] :[font = input; preserveAspect; startGroup] DSolve[ ode,y,x ] :[font = message; inactive; preserveAspect] DSolve::dnim: Built-in procedures cannot solve this differential equation. :[font = output; output; inactive; preserveAspect; endGroup; endGroup] DSolve[f[x*Derivative[1][y][x]] == g[Derivative[1][y][x]], y, x] ;[o] DSolve[f[x y'[x]] == g[y'[x]], y, x] :[font = section; inactive; preserveAspect; startGroup] Equation (9) :[font = input; preserveAspect; startGroup] ode = y'[x] == (3 x^2 - y[x]^2 - 7)/(Exp[y[x]]+2 x y[x]+1) :[font = output; output; inactive; preserveAspect; endGroup] Derivative[1][y][x] == (-7 + 3*x^2 - y[x]^2)/ (1 + E^y[x] + 2*x*y[x]) ;[o] 2 2 -7 + 3 x - y[x] y'[x] == -------------------- y[x] 1 + E + 2 x y[x] :[font = input; preserveAspect; startGroup] DSolve[ ode,y,x ] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] DSolve[Derivative[1][y][x] == (-7 + 3*x^2 - y[x]^2)/(1 + E^y[x] + 2*x*y[x]), y, x] ;[o] 2 2 -7 + 3 x - y[x] DSolve[y'[x] == --------------------, y, x] y[x] 1 + E + 2 x y[x] :[font = section; inactive; preserveAspect; startGroup] Equation (10) :[font = input; preserveAspect; startGroup] ode = y'[x] == (2 x^3 y[x] - y[x]^4) / (x^4 - 2 x y[x]^3) :[font = output; output; inactive; preserveAspect; endGroup] Derivative[1][y][x] == (2*x^3*y[x] - y[x]^4)/ (x^4 - 2*x*y[x]^3) ;[o] 3 4 2 x y[x] - y[x] y'[x] == ----------------- 4 3 x - 2 x y[x] :[font = input; preserveAspect; startGroup] DSolve[ ode,y,x ] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] DSolve[Derivative[1][y][x] == (2*x^3*y[x] - y[x]^4)/(x^4 - 2*x*y[x]^3), y, x] ;[o] 3 4 2 x y[x] - y[x] DSolve[y'[x] == -----------------, y, x] 4 3 x - 2 x y[x] :[font = section; inactive; preserveAspect; startGroup] Equation (11) :[font = input; preserveAspect; startGroup] ode = y'[x] (y'[x] + y[x]) == x (x + y[x]) :[font = output; output; inactive; preserveAspect; endGroup] Derivative[1][y][x]*(y[x] + Derivative[1][y][x]) == x*(x + y[x]) ;[o] y'[x] (y[x] + y'[x]) == x (x + y[x]) :[font = input; preserveAspect; startGroup] DSolve[ ode,y,x ] :[font = message; inactive; preserveAspect] DSolve::dnim: Built-in procedures cannot solve this differential equation. :[font = output; output; inactive; preserveAspect; endGroup] DSolve[Derivative[1][y][x]*(y[x] + Derivative[1][y][x]) == x*(x + y[x]), y, x] ;[o] DSolve[y'[x] (y[x] + y'[x]) == x (x + y[x]), y, x] :[font = text; inactive; preserveAspect; plain; bold] Factored form of ode11: :[font = input; preserveAspect; startGroup] odeFactored = (y'[x] + y[x] + x)(y'[x] - x) == 0 :[font = output; output; inactive; preserveAspect; endGroup] (-x + Derivative[1][y][x])* (x + y[x] + Derivative[1][y][x]) == 0 ;[o] (-x + y'[x]) (x + y[x] + y'[x]) == 0 :[font = input; preserveAspect; startGroup] DSolve[ odeFactored,y,x ] :[font = message; inactive; preserveAspect] DSolve::dnim: Built-in procedures cannot solve this differential equation. :[font = output; output; inactive; preserveAspect; endGroup; endGroup] DSolve[(-x + Derivative[1][y][x])* (x + y[x] + Derivative[1][y][x]) == 0, y, x] ;[o] DSolve[(-x + y'[x]) (x + y[x] + y'[x]) == 0, y, x] :[font = section; inactive; preserveAspect; startGroup] Equation (12) :[font = input; preserveAspect; startGroup] ode = y'[x] == x/(x^2 y[x]^2 + y[x]^5) :[font = output; output; inactive; preserveAspect; endGroup] Derivative[1][y][x] == x/(x^2*y[x]^2 + y[x]^5) ;[o] x y'[x] == ---------------- 2 2 5 x y[x] + y[x] :[font = input; preserveAspect; startGroup] DSolve[ ode,y,x ] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] DSolve[Derivative[1][y][x] == x/(x^2*y[x]^2 + y[x]^5), y, x] ;[o] x DSolve[y'[x] == ----------------, y, x] 2 2 5 x y[x] + y[x] :[font = section; inactive; preserveAspect; startGroup] Equation (13) :[font = input; preserveAspect; startGroup] ode = y[x] == 2 x y'[x] - a y'[x]^3 :[font = output; output; inactive; preserveAspect; endGroup] y[x] == 2*x*Derivative[1][y][x] - a*Derivative[1][y][x]^3 ;[o] 3 y[x] == 2 x y'[x] - a y'[x] :[font = input; preserveAspect; startGroup] DSolve[ ode,y,x ] :[font = message; inactive; preserveAspect] DSolve::dnim: Built-in procedures cannot solve this differential equation. :[font = output; output; inactive; preserveAspect; endGroup; endGroup] DSolve[y[x] == 2*x*Derivative[1][y][x] - a*Derivative[1][y][x]^3, y, x] ;[o] 3 DSolve[y[x] == 2 x y'[x] - a y'[x] , y, x] :[font = section; inactive; preserveAspect; startGroup] Equation (14) :[font = input; preserveAspect; startGroup] ode = y[x] == 2 x y'[x] - y'[x]^2 :[font = output; output; inactive; preserveAspect; endGroup] y[x] == 2*x*Derivative[1][y][x] - Derivative[1][y][x]^2 ;[o] 2 y[x] == 2 x y'[x] - y'[x] :[font = input; preserveAspect; startGroup] DSolve[ ode,y,x ] :[font = message; inactive; preserveAspect] DSolve::dnim: Built-in procedures cannot solve this differential equation. :[font = output; output; inactive; preserveAspect; endGroup; endGroup] DSolve[y[x] == 2*x*Derivative[1][y][x] - Derivative[1][y][x]^2, y, x] ;[o] 2 DSolve[y[x] == 2 x y'[x] - y'[x] , y, x] :[font = section; inactive; preserveAspect; startGroup] Equation (15) :[font = input; preserveAspect; startGroup] ode = y'[x] == Exp[x] y[x]^2 - y[x] + Exp[-x] :[font = output; output; inactive; preserveAspect; endGroup] Derivative[1][y][x] == E^(-x) - y[x] + E^x*y[x]^2 ;[o] -x x 2 y'[x] == E - y[x] + E y[x] :[font = input; preserveAspect; startGroup] DSolve[ ode,y,x ] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] DSolve[Derivative[1][y][x] == E^(-x) - y[x] + E^x*y[x]^2, y, x] ;[o] -x x 2 DSolve[y'[x] == E - y[x] + E y[x] , y, x] :[font = section; inactive; preserveAspect; startGroup] Equation (16) :[font = input; preserveAspect; startGroup] ode = y'[x] == y[x]^2 - x y[x] + 1 :[font = output; output; inactive; preserveAspect; endGroup] Derivative[1][y][x] == 1 - x*y[x] + y[x]^2 ;[o] 2 y'[x] == 1 - x y[x] + y[x] :[font = input; preserveAspect; startGroup] DSolve[ ode,y,x ] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] DSolve[Derivative[1][y][x] == 1 - x*y[x] + y[x]^2, y, x] ;[o] 2 DSolve[y'[x] == 1 - x y[x] + y[x] , y, x] :[font = section; inactive; preserveAspect; startGroup] Equation (17) :[font = input; preserveAspect; startGroup] ode = y'[x] == (9 x^8 + 1)/(y[x]^2+1) :[font = output; output; inactive; preserveAspect; endGroup] Derivative[1][y][x] == (1 + 9*x^8)/(1 + y[x]^2) ;[o] 8 1 + 9 x y'[x] == --------- 2 1 + y[x] :[font = input; preserveAspect; startGroup] DSolve[ ode,y,x ] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] {{y -> Function[x, (-3*2^(1/3))/ (81*(x + x^9 + C[1]) + (2916 + 6561*(x + x^9 + C[1])^2)^(1/2))^(1/3) + (81*(x + x^9 + C[1]) + (2916 + 6561*(x + x^9 + C[1])^2)^(1/2))^(1/3)/ (3*2^(1/3))]}, {y -> Function[x, (3*(1 + I*3^(1/2)))/ (2^(2/3)*(81*(x + x^9 + C[1]) + (2916 + 6561*(x + x^9 + C[1])^2)^(1/2))^(1/3)) \ - ((1 - I*3^(1/2))* (81*(x + x^9 + C[1]) + (2916 + 6561*(x + x^9 + C[1])^2)^(1/2))^(1/3))/ (6*2^(1/3))]}, {y -> Function[x, (3*(1 - I*3^(1/2)))/ (2^(2/3)*(81*(x + x^9 + C[1]) + (2916 + 6561*(x + x^9 + C[1])^2)^(1/2))^(1/3)) \ - ((1 + I*3^(1/2))* (81*(x + x^9 + C[1]) + (2916 + 6561*(x + x^9 + C[1])^2)^(1/2))^(1/3))/ (6*2^(1/3))]}} ;[o] 1/3 {{y -> Function[x, (-3 2 ) / 9 Power[81 (x + x + C[1]) + 9 2 Sqrt[2916 + 6561 (x + x + C[1]) ], 1/3] + 9 Power[81 (x + x + C[1]) + 9 2 1/3 Sqrt[2916 + 6561 (x + x + C[1]) ], 1/3] / (3 2 ) ]}, {y -> Function[x, (3 (1 + I Sqrt[3])) / 2/3 9 (2 Power[81 (x + x + C[1]) + 9 2 Sqrt[2916 + 6561 (x + x + C[1]) ], 1/3]) - 9 ((1 - I Sqrt[3]) Power[81 (x + x + C[1]) + 9 2 Sqrt[2916 + 6561 (x + x + C[1]) ], 1/3]) / 1/3 (6 2 )]}, {y -> Function[x, (3 (1 - I Sqrt[3])) / 2/3 9 (2 Power[81 (x + x + C[1]) + 9 2 Sqrt[2916 + 6561 (x + x + C[1]) ], 1/3]) - 9 ((1 + I Sqrt[3]) Power[81 (x + x + C[1]) + 9 2 Sqrt[2916 + 6561 (x + x + C[1]) ], 1/3]) / 1/3 (6 2 )]}} :[font = section; inactive; preserveAspect; startGroup] Equation (18) :[font = input; preserveAspect; startGroup] ode = y[x] == 2 x y'[x] + y[x] y'[x]^2 :[font = output; output; inactive; preserveAspect; endGroup] y[x] == 2*x*Derivative[1][y][x] + y[x]*Derivative[1][y][x]^2 ;[o] 2 y[x] == 2 x y'[x] + y[x] y'[x] :[font = input; preserveAspect; startGroup] DSolve[ ode,y,x ] :[font = message; inactive; preserveAspect] DSolve::dnim: Built-in procedures cannot solve this differential equation. :[font = output; output; inactive; preserveAspect; endGroup; endGroup] DSolve[y[x] == 2*x*Derivative[1][y][x] + y[x]*Derivative[1][y][x]^2, y, x] ;[o] 2 DSolve[y[x] == 2 x y'[x] + y[x] y'[x] , y, x] :[font = section; inactive; preserveAspect; startGroup] Equation (19) :[font = input; preserveAspect; startGroup] ode = y[x] y'[x] - x y'[x]^2 == x :[font = output; output; inactive; preserveAspect; endGroup] y[x]*Derivative[1][y][x] - x*Derivative[1][y][x]^2 == x ;[o] 2 y[x] y'[x] - x y'[x] == x :[font = input; preserveAspect; startGroup] DSolve[ ode,y,x ] :[font = message; inactive; preserveAspect] DSolve::dnim: Built-in procedures cannot solve this differential equation. :[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup] DSolve[y[x]*Derivative[1][y][x] - x*Derivative[1][y][x]^2 == x, y, x] ;[o] 2 DSolve[y[x] y'[x] - x y'[x] == x, y, x] :[font = title; inactive; preserveAspect; startGroup] Second order equations ;[s] 1:0,0;22,-1; 1:1,0,0 ,times,1,18,0,0,0; :[font = section; inactive; preserveAspect; startGroup] Equation (20) :[font = input; preserveAspect; startGroup] ode = y''[x] (a x + b)^2 + 4 y'[x] (a x + b) a + 2 y[x] a^2 == 0 :[font = output; output; inactive; preserveAspect; endGroup] 2*a^2*y[x] + 4*a*(b + a*x)*Derivative[1][y][x] + (b + a*x)^2*Derivative[2][y][x] == 0 ;[o] 2 2 2 a y[x] + 4 a (b + a x) y'[x] + (b + a x) y''[x] == 0 :[font = input; preserveAspect; startGroup] DSolve[ ode,y,x ] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] {{y -> Function[x, (x*C[1] - C[2])/(b + a*x)^2]}} ;[o] x C[1] - C[2] {{y -> Function[x, -------------]}} 2 (b + a x) :[font = section; inactive; preserveAspect; startGroup] Equation (21) :[font = input; preserveAspect; startGroup] ode = (x^2-x) u''[x] + (2 x^2+4 x -3) u'[x] + 8 x u[x] == 1 :[font = output; output; inactive; preserveAspect; endGroup] 8*x*u[x] + (-3 + 4*x + 2*x^2)*Derivative[1][u][x] + (-x + x^2)*Derivative[2][u][x] == 1 ;[o] 2 2 8 x u[x] + (-3 + 4 x + 2 x ) u'[x] + (-x + x ) u''[x] == 1 :[font = input; preserveAspect; startGroup] DSolve[ ode,u,x ] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] DSolve[8*x*u[x] + (-3 + 4*x + 2*x^2)*Derivative[1][u][x] + (-x + x^2)*Derivative[2][u][x] == 1, u, x] ;[o] 2 DSolve[8 x u[x] + (-3 + 4 x + 2 x ) u'[x] + 2 (-x + x ) u''[x] == 1, u, x] :[font = section; inactive; preserveAspect; startGroup] Equation (22) :[font = input; preserveAspect; startGroup] ode = (x^2-x) w''[x] + (1-2 x^2) w'[x] + (4 x-2) w[x] == 0 :[font = output; output; inactive; preserveAspect; endGroup] (-2 + 4*x)*w[x] + (1 - 2*x^2)*Derivative[1][w][x] + (-x + x^2)*Derivative[2][w][x] == 0 ;[o] 2 2 (-2 + 4 x) w[x] + (1 - 2 x ) w'[x] + (-x + x ) w''[x] == 0 :[font = input; preserveAspect; startGroup] DSolve[ ode,w,x ] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] DSolve[(-2 + 4*x)*w[x] + (1 - 2*x^2)*Derivative[1][w][x] + (-x + x^2)*Derivative[2][w][x] == 0, w, x] ;[o] 2 DSolve[(-2 + 4 x) w[x] + (1 - 2 x ) w'[x] + 2 (-x + x ) w''[x] == 0, w, x] :[font = section; inactive; preserveAspect; startGroup] Equation (23) :[font = input; preserveAspect; startGroup] ode = y''[x] - y'[x] == 2 y[x] y'[x] :[font = output; output; inactive; preserveAspect; endGroup] -Derivative[1][y][x] + Derivative[2][y][x] == 2*y[x]*Derivative[1][y][x] ;[o] -y'[x] + y''[x] == 2 y[x] y'[x] :[font = input; preserveAspect; startGroup] DSolve[ ode,y,x ] :[font = message; inactive; preserveAspect] DSolve::dnim: Built-in procedures cannot solve this differential equation. :[font = output; output; inactive; preserveAspect; endGroup; endGroup] DSolve[-Derivative[1][y][x] + Derivative[2][y][x] == 2*y[x]*Derivative[1][y][x], y, x] ;[o] DSolve[-y'[x] + y''[x] == 2 y[x] y'[x], y, x] :[font = section; inactive; preserveAspect; startGroup] Equation (24) :[font = input; preserveAspect; startGroup] ode = y''[x]/y[x] - y'[x]^2/y[x]^2 - 1 + 1/y[x]^3 == 0 :[font = output; output; inactive; preserveAspect; endGroup] -1 + y[x]^(-3) - Derivative[1][y][x]^2/y[x]^2 + Derivative[2][y][x]/y[x] == 0 ;[o] 2 -3 y'[x] y''[x] -1 + y[x] - ------ + ------ == 0 2 y[x] y[x] :[font = input; preserveAspect; startGroup] DSolve[ ode,y,x ] :[font = message; inactive; preserveAspect] DSolve::dnim: Built-in procedures cannot solve this differential equation. :[font = output; output; inactive; preserveAspect; endGroup; endGroup] DSolve[-1 + y[x]^(-3) - Derivative[1][y][x]^2/y[x]^2 + Derivative[2][y][x]/y[x] == 0, y, x] ;[o] 2 -3 y'[x] y''[x] DSolve[-1 + y[x] - ------ + ------ == 0, y, x] 2 y[x] y[x] :[font = section; inactive; preserveAspect; startGroup] Equation (25) :[font = input; preserveAspect; startGroup] ode = y''[x] + 2 x y'[x] == 2 x :[font = output; output; inactive; preserveAspect; endGroup] 2*x*Derivative[1][y][x] + Derivative[2][y][x] == 2*x ;[o] 2 x y'[x] + y''[x] == 2 x :[font = input; preserveAspect; startGroup] DSolve[ ode,y,x ] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] DSolve[2*x*Derivative[1][y][x] + Derivative[2][y][x] == 2*x, y, x] ;[o] DSolve[2 x y'[x] + y''[x] == 2 x, y, x] :[font = section; inactive; preserveAspect; startGroup] Equation (26) :[font = input; preserveAspect; startGroup] ode = 2 y[x] y''[x] - y'[x]^2 == 1/3 (y'[x] - x y''[x])^2 :[font = output; output; inactive; preserveAspect; endGroup] -Derivative[1][y][x]^2 + 2*y[x]*Derivative[2][y][x] == (Derivative[1][y][x] - x*Derivative[2][y][x])^2/3 ;[o] 2 2 (y'[x] - x y''[x]) -y'[x] + 2 y[x] y''[x] == ------------------- 3 :[font = input; preserveAspect; startGroup] DSolve[ ode,y,x ] :[font = message; inactive; preserveAspect] DSolve::dnim: Built-in procedures cannot solve this differential equation. :[font = output; output; inactive; preserveAspect; endGroup; endGroup] DSolve[-Derivative[1][y][x]^2 + 2*y[x]*Derivative[2][y][x] == (Derivative[1][y][x] - x*Derivative[2][y][x])^2/3, y, x] ;[o] 2 2 (y'[x] - x y''[x]) DSolve[-y'[x] + 2 y[x] y''[x] == -------------------, y, x] 3 :[font = section; inactive; preserveAspect; startGroup] Equation (27) :[font = input; preserveAspect; startGroup] ode = x y''[x] == 2 y[x] y'[x] :[font = output; output; inactive; preserveAspect; endGroup] x*Derivative[2][y][x] == 2*y[x]*Derivative[1][y][x] ;[o] x y''[x] == 2 y[x] y'[x] :[font = input; preserveAspect; startGroup] DSolve[ ode,y,x ] :[font = message; inactive; preserveAspect] DSolve::dnim: Built-in procedures cannot solve this differential equation. :[font = output; output; inactive; preserveAspect; endGroup; endGroup] DSolve[x*Derivative[2][y][x] == 2*y[x]*Derivative[1][y][x], y, x] ;[o] DSolve[x y''[x] == 2 y[x] y'[x], y, x] :[font = section; inactive; preserveAspect; startGroup] Equation (28) :[font = input; preserveAspect; startGroup] ode = (1-x)(y[x] y''[x] - y'[x]^2) + x^2 y[x]^2 == 0 :[font = output; output; inactive; preserveAspect; endGroup] x^2*y[x]^2 + (1 - x)*(-Derivative[1][y][x]^2 + y[x]*Derivative[2][y][x]) == 0 ;[o] 2 2 2 x y[x] + (1 - x) (-y'[x] + y[x] y''[x]) == 0 :[font = input; preserveAspect; startGroup] DSolve[ ode,y,x ] :[font = message; inactive; preserveAspect] DSolve::dnim: Built-in procedures cannot solve this differential equation. :[font = output; output; inactive; preserveAspect; endGroup; endGroup] DSolve[x^2*y[x]^2 + (1 - x)* (-Derivative[1][y][x]^2 + y[x]*Derivative[2][y][x]) == 0, y, x] ;[o] 2 2 2 DSolve[x y[x] + (1 - x) (-y'[x] + y[x] y''[x]) == 0, y, x] :[font = section; inactive; preserveAspect; startGroup] Equation (29) :[font = input; preserveAspect; startGroup] ode = x y[x] y''[x] + x y'[x]^2 + y[x] y'[x] == 0 :[font = output; output; inactive; preserveAspect; endGroup] y[x]*Derivative[1][y][x] + x*Derivative[1][y][x]^2 + x*y[x]*Derivative[2][y][x] == 0 ;[o] 2 y[x] y'[x] + x y'[x] + x y[x] y''[x] == 0 :[font = input; preserveAspect; startGroup] DSolve[ ode,y,x ] :[font = message; inactive; preserveAspect] DSolve::dnim: Built-in procedures cannot solve this differential equation. :[font = output; output; inactive; preserveAspect; endGroup; endGroup] DSolve[y[x]*Derivative[1][y][x] + x*Derivative[1][y][x]^2 + x*y[x]*Derivative[2][y][x] == 0, y, x] ;[o] 2 DSolve[y[x] y'[x] + x y'[x] + x y[x] y''[x] == 0, y, x] :[font = section; inactive; preserveAspect; startGroup] Equation (30) :[font = input; preserveAspect; startGroup] ode = y''[x]^2 - 2 y'[x] y''[x] - 2 y[x] y'[x] - y[x]^2 == 0 :[font = output; output; inactive; preserveAspect; endGroup] changed ;[o] changed :[font = input; preserveAspect; startGroup] DSolve[ ode,y,x ] :[font = message; inactive; preserveAspect] changed :[font = output; output; inactive; preserveAspect; endGroup; endGroup] changed :[font = section; inactive; preserveAspect; startGroup] Equation (31) :[font = input; preserveAspect; startGroup] ode = (x^3/2 - x^2) y''[x] + (2 x^2 - 3 x + 1) y'[x] + (x-1) y[x] == 0 :[font = output; output; inactive; preserveAspect; endGroup] (-1 + x)*y[x] + (1 - 3*x + 2*x^2)*Derivative[1][y][x] + (-x^2 + x^3/2)*Derivative[2][y][x] == 0 ;[o] 2 (-1 + x) y[x] + (1 - 3 x + 2 x ) y'[x] + 3 2 x (-x + --) y''[x] == 0 2 :[font = input; preserveAspect; startGroup] DSolve[ ode,y,x ] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] {{y -> Function[x, (C[1] - ((E*Pi)/2)^(1/2)*C[2]* Erfi[(2 - x)^(1/2)/(2^(1/2)*x^(1/2))])/ (E^x^(-1)*(2 - x)^(1/2)*x^(1/2))]}} ;[o] {{y -> Function[x, E Pi Sqrt[2 - x] C[1] - Sqrt[----] C[2] Erfi[---------------] 2 Sqrt[2] Sqrt[x] --------------------------------------------]}} 1/x E Sqrt[2 - x] Sqrt[x] :[font = section; inactive; preserveAspect; startGroup] Equation (32) :[font = input; preserveAspect; startGroup] ode = y''[x] - 2 x y'[x] + 2 y[x] == 3 :[font = output; output; inactive; preserveAspect; endGroup] 2*y[x] - 2*x*Derivative[1][y][x] + Derivative[2][y][x] == 3 ;[o] 2 y[x] - 2 x y'[x] + y''[x] == 3 :[font = input; preserveAspect; startGroup] DSolve[ ode,y,x ] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] DSolve[2*y[x] - 2*x*Derivative[1][y][x] + Derivative[2][y][x] == 3, y, x] ;[o] DSolve[2 y[x] - 2 x y'[x] + y''[x] == 3, y, x] :[font = section; inactive; preserveAspect; startGroup] Equation (33) :[font = input; preserveAspect; startGroup] ode = Sqrt[x] y''[x] + 2 x y'[x] + 3 y[x] == 0 :[font = output; output; inactive; preserveAspect; endGroup] 3*y[x] + 2*x*Derivative[1][y][x] + x^(1/2)*Derivative[2][y][x] == 0 ;[o] 3 y[x] + 2 x y'[x] + Sqrt[x] y''[x] == 0 :[font = input; preserveAspect; startGroup] DSolve[ ode,y,x ] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] DSolve[3*y[x] + 2*x*Derivative[1][y][x] + x^(1/2)*Derivative[2][y][x] == 0, y, x] ;[o] DSolve[3 y[x] + 2 x y'[x] + Sqrt[x] y''[x] == 0, y, x] :[font = section; inactive; preserveAspect; startGroup] Equation (34) :[font = input; preserveAspect; startGroup] ode = x^2 y''[x] + 3 x y'[x] == 1/(y[x]^3 x^4) :[font = output; output; inactive; preserveAspect; endGroup] 3*x*Derivative[1][y][x] + x^2*Derivative[2][y][x] == 1/(x^4*y[x]^3) ;[o] 2 1 3 x y'[x] + x y''[x] == -------- 4 3 x y[x] :[font = input; preserveAspect; startGroup] DSolve[ ode,y,x ] :[font = message; inactive; preserveAspect] DSolve::dnim: Built-in procedures cannot solve this differential equation. :[font = output; output; inactive; preserveAspect; endGroup; endGroup] DSolve[3*x*Derivative[1][y][x] + x^2*Derivative[2][y][x] == 1/(x^4*y[x]^3), y, x] ;[o] 2 1 DSolve[3 x y'[x] + x y''[x] == --------, y, x] 4 3 x y[x] :[font = section; inactive; preserveAspect; startGroup] Equation (35) :[font = input; preserveAspect; startGroup] ode = y''[x] - 2/x^2 y[x] == 7 x^4 + 3 x^3 :[font = output; output; inactive; preserveAspect; endGroup] (-2*y[x])/x^2 + Derivative[2][y][x] == 3*x^3 + 7*x^4 ;[o] -2 y[x] 3 4 ------- + y''[x] == 3 x + 7 x 2 x :[font = input; preserveAspect; startGroup] DSolve[ ode,y,x ] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] DSolve[(-2*y[x])/x^2 + Derivative[2][y][x] == 3*x^3 + 7*x^4, y, x] ;[o] -2 y[x] 3 4 DSolve[------- + y''[x] == 3 x + 7 x , y, x] 2 x :[font = section; inactive; preserveAspect; startGroup] Equation (36) :[font = input; preserveAspect; startGroup] ode = y''[x] + y[x] == Csc[x] :[font = output; output; inactive; preserveAspect; endGroup] y[x] + Derivative[2][y][x] == Csc[x] ;[o] y[x] + y''[x] == Csc[x] :[font = input; preserveAspect; startGroup] DSolve[ ode,y[x],x ] :[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup] {{y[x] -> C[2]*Cos[x] - x*(Cos[x] + I*Sin[x]) - C[1]*Sin[x] + I/2*Log[-1 + Cos[2*x] + I*Sin[2*x]]* (Cos[x] + I*Sin[x])*(-1 + Cos[2*x] - I*Sin[2*x])}} ;[o] {{y[x] -> C[2] Cos[x] - x (Cos[x] + I Sin[x]) - I C[1] Sin[x] + - Log[-1 + Cos[2 x] + I Sin[2 x]] 2 (Cos[x] + I Sin[x]) (-1 + Cos[2 x] - I Sin[2 x])}} :[font = title; inactive; preserveAspect; startGroup] Higher order equations ;[s] 1:0,0;22,-1; 1:1,0,0 ,times,1,18,0,0,0; :[font = section; inactive; preserveAspect; startGroup] Equation (37) :[font = input; preserveAspect; startGroup] ode = y'''''''[x]-14 y''''''[x] + 80 y'''''[x] - 242 y''''[x] + 419 y'''[x] - 416 y''[x] + 220 y'[x] - 48 y[x] == 0 :[font = output; output; inactive; preserveAspect; endGroup] -48*y[x] + 220*Derivative[1][y][x] - 416*Derivative[2][y][x] + 419*Derivative[3][y][x] - 242*Derivative[4][y][x] + 80*Derivative[5][y][x] - 14*Derivative[6][y][x] + Derivative[7][y][x] == 0 ;[o] (3) -48 y[x] + 220 y'[x] - 416 y''[x] + 419 y [x] - (4) (5) (6) (7) 242 y [x] + 80 y [x] - 14 y [x] + y [x] == 0 :[font = input; preserveAspect; startGroup] DSolve[ ode,y,x ] :[font = message; inactive; preserveAspect] DSolve::dsdeg: Warning: Differential equation of order higher than four encountered. DSolve may not be able to find the solution. :[font = output; output; inactive; preserveAspect; endGroup; endGroup] {{y -> Function[x, E^x*C[1] + E^x*x*C[2] + E^x*x^2*C[3] + E^(2*x)*C[4] + E^(2*x)*x*C[5] + E^(3*x)*C[6] + E^(4*x)*C[7]]}} ;[o] x x x 2 {{y -> Function[x, E C[1] + E x C[2] + E x C[3] + 2 x 2 x 3 x 4 x E C[4] + E x C[5] + E C[6] + E C[7]]}} :[font = section; inactive; preserveAspect; startGroup] Equation (38) :[font = input; preserveAspect; startGroup] ode = y''''[x] - 4/x^2 y''[x] + 8/x^3 y'[x] - 8/x^4 y[x] == 0 :[font = output; output; inactive; preserveAspect; endGroup] (-8*y[x])/x^4 + (8*Derivative[1][y][x])/x^3 - (4*Derivative[2][y][x])/x^2 + Derivative[4][y][x] == 0 ;[o] -8 y[x] 8 y'[x] 4 y''[x] (4) ------- + ------- - -------- + y [x] == 0 4 3 2 x x x :[font = input; preserveAspect; startGroup] DSolve[ ode,y,x ] :[font = message; inactive; preserveAspect] DSolve::dnim: Built-in procedures cannot solve this differential equation. :[font = output; output; inactive; preserveAspect; endGroup; endGroup] DSolve[(-8*y[x])/x^4 + (8*Derivative[1][y][x])/x^3 - (4*Derivative[2][y][x])/x^2 + Derivative[4][y][x] == 0, y, x] ;[o] -8 y[x] 8 y'[x] 4 y''[x] (4) DSolve[------- + ------- - -------- + y [x] == 0, y, x] 4 3 2 x x x :[font = section; inactive; preserveAspect; startGroup] Equation (39) :[font = input; preserveAspect; startGroup] ode = (1+x+x^2) y'''[x] + (3 + 6 x) y''[x] + 6 y'[x] == 6 x :[font = output; output; inactive; preserveAspect; endGroup] 6*Derivative[1][y][x] + (3 + 6*x)*Derivative[2][y][x] + (1 + x + x^2)*Derivative[3][y][x] == 6*x ;[o] 2 (3) 6 y'[x] + (3 + 6 x) y''[x] + (1 + x + x ) y [x] == 6 x :[font = input; preserveAspect; startGroup] DSolve[ ode,y,x ] :[font = message; inactive; preserveAspect] DSolve::dnim: Built-in procedures cannot solve this differential equation. :[font = output; output; inactive; preserveAspect; endGroup; endGroup] DSolve[6*Derivative[1][y][x] + (3 + 6*x)*Derivative[2][y][x] + (1 + x + x^2)*Derivative[3][y][x] == 6*x, y, x] ;[o] 2 (3) DSolve[6 y'[x] + (3 + 6 x) y''[x] + (1 + x + x ) y [x] == 6 x, y, x] :[font = section; inactive; preserveAspect; startGroup] Equation (40) :[font = input; preserveAspect; startGroup] ode = (y'[x]^2+1) y'''[x] - 3 y'[x] y''[x]^2 == 0 :[font = output; output; inactive; preserveAspect; endGroup] -3*Derivative[1][y][x]*Derivative[2][y][x]^2 + (1 + Derivative[1][y][x]^2)*Derivative[3][y][x] == 0 ;[o] 2 2 (3) -3 y'[x] y''[x] + (1 + y'[x] ) y [x] == 0 :[font = input; preserveAspect; startGroup] DSolve[ ode,y,x ] :[font = message; inactive; preserveAspect] DSolve::dnim: Built-in procedures cannot solve this differential equation. :[font = output; output; inactive; preserveAspect; endGroup; endGroup] DSolve[-3*Derivative[1][y][x]*Derivative[2][y][x]^2 + (1 + Derivative[1][y][x]^2)*Derivative[3][y][x] == 0, y, x] ;[o] 2 2 (3) DSolve[-3 y'[x] y''[x] + (1 + y'[x] ) y [x] == 0, y, x] :[font = section; inactive; preserveAspect; startGroup] Equation (41) :[font = input; preserveAspect; startGroup] ode = 3 y''[x] y''''[x] - 5 y'''[x]^2 == 0 :[font = output; output; inactive; preserveAspect; endGroup] -5*Derivative[3][y][x]^2 + 3*Derivative[2][y][x]*Derivative[4][y][x] == 0 ;[o] (3) 2 (4) -5 y [x] + 3 y''[x] y [x] == 0 :[font = input; preserveAspect; startGroup] DSolve[ ode,y,x ] :[font = message; inactive; preserveAspect] DSolve::dnim: Built-in procedures cannot solve this differential equation. :[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup] DSolve[-5*Derivative[3][y][x]^2 + 3*Derivative[2][y][x]*Derivative[4][y][x] == 0, y, x] ;[o] (3) 2 (4) DSolve[-5 y [x] + 3 y''[x] y [x] == 0, y, x] :[font = title; inactive; preserveAspect; startGroup] Special equations ;[s] 1:0,0;17,-1; 1:1,0,0 ,times,1,18,0,0,0; :[font = section; inactive; preserveAspect; startGroup] Equation (42) :[font = input; preserveAspect; startGroup] ode = y'[t] + a y[t-1] == 0 :[font = output; output; inactive; preserveAspect; endGroup] a*y[-1 + t] + Derivative[1][y][t] == 0 ;[o] a y[-1 + t] + y'[t] == 0 :[font = input; preserveAspect; startGroup] DSolve[ ode,y,t ] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] {{y -> Function[t, C[1] - a*Integrate[y[-1 + DSolve`t], {DSolve`t, 0, t}]]}} ;[o] {{y -> Function[t, C[1] - a Integrate[y[-1 + DSolve`t], {DSolve`t, 0, t}]]}} :[font = section; inactive; preserveAspect; startGroup] Equation (43) :[font = input; preserveAspect; startGroup] ode = D[y[x,a],x] == a y[x,a] :[font = output; output; inactive; preserveAspect; endGroup] Derivative[1, 0][y][x, a] == a*y[x, a] ;[o] (1,0) y [x, a] == a y[x, a] :[font = input; preserveAspect; startGroup] DSolve[ ode,y,x ] :[font = message; inactive; preserveAspect] DSolve::deqx: Supplied equations are not differential equations of the given functions. :[font = output; output; inactive; preserveAspect; endGroup; endGroup] DSolve[Derivative[1, 0][y][x, a] == a*y[x, a], y, x] ;[o] (1,0) DSolve[y [x, a] == a y[x, a], y, x] :[font = subtitle; inactive; preserveAspect; startGroup] Single equations with initial conditions ;[s] 1:0,0;40,-1; 1:1,0,0 ,times,1,24,0,0,0; :[font = section; inactive; preserveAspect; startGroup] Equation (44) :[font = input; preserveAspect; startGroup] ode = y''''[x] == Sin[x] :[font = output; output; inactive; preserveAspect; endGroup] Derivative[4][y][x] == Sin[x] ;[o] (4) y [x] == Sin[x] :[font = input; preserveAspect; startGroup] DSolve[ {ode, y[0]==0,y'[0]==0,y''[0]==0,y'''[0]==0},y,x ] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] {{y -> Function[x, -x + x^3/6 + Sin[x]]}} ;[o] 3 x {{y -> Function[x, -x + -- + Sin[x]]}} 6 :[font = section; inactive; preserveAspect; startGroup] Equation (45) :[font = input; preserveAspect; startGroup] ode = x y''[x] + y'[x] + 2 x y[x] == 0 :[font = output; output; inactive; preserveAspect; endGroup] 2*x*y[x] + Derivative[1][y][x] + x*Derivative[2][y][x] == 0 ;[o] 2 x y[x] + y'[x] + x y''[x] == 0 :[font = input; preserveAspect; startGroup] DSolve[ {ode, y[0]==1,y'[0]==0},y,x ] :[font = message; inactive; preserveAspect] Infinity::indet: Indeterminate expression ComplexInfinity + <<1>> encountered. :[font = message; inactive; preserveAspect] Solve::svars: Warning: Equations may not give solutions for all "solve" variables. :[font = output; output; inactive; preserveAspect; endGroup; endGroup] {{y -> Function[x, BesselY[0, 2^(1/2)*x]*C[2] + BesselJ[0, 2^(1/2)*x]*(1 + C[2]*DirectedInfinity[1])] }} ;[o] {{y -> Function[x, BesselY[0, Sqrt[2] x] C[2] + BesselJ[0, Sqrt[2] x] (1 + C[2] (Infinity))]}} :[font = section; inactive; preserveAspect; startGroup] Equation (46) :[font = input; preserveAspect; startGroup] ode = x y'[x]^2 - y[x]^2 + 1 == 0 :[font = output; output; inactive; preserveAspect; endGroup] 1 - y[x]^2 + x*Derivative[1][y][x]^2 == 0 ;[o] 2 2 1 - y[x] + x y'[x] == 0 :[font = input; preserveAspect; startGroup] DSolve[ {ode, y[0]==1},y,x ] :[font = message; inactive; preserveAspect] DSolve::dnim: Built-in procedures cannot solve this differential equation. :[font = output; output; inactive; preserveAspect; endGroup; endGroup] DSolve[{1 - y[x]^2 + x*Derivative[1][y][x]^2 == 0, y[0] == 1}, y, x] ;[o] 2 2 DSolve[{1 - y[x] + x y'[x] == 0, y[0] == 1}, y, x] :[font = section; inactive; preserveAspect; startGroup] Equation (47) :[font = input; preserveAspect; startGroup] ode = y''[x] + y[x] y'[x]^3 == 0 :[font = output; output; inactive; preserveAspect; endGroup] y[x]*Derivative[1][y][x]^3 + Derivative[2][y][x] == 0 ;[o] 3 y[x] y'[x] + y''[x] == 0 :[font = input; preserveAspect; startGroup] DSolve[ {ode, y[0]==0, y'[0]==2},y,x ] :[font = message; inactive; preserveAspect] DSolve::dnim: Built-in procedures cannot solve this differential equation. :[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup; endGroup] DSolve[{y[x]*Derivative[1][y][x]^3 + Derivative[2][y][x] == 0, y[0] == 0, Derivative[1][y][0] == 2}, y, x] ;[o] 3 DSolve[{y[x] y'[x] + y''[x] == 0, y[0] == 0, y'[0] == 2}, y, x] :[font = title; inactive; preserveAspect; startGroup] Systems of equations :[font = section; inactive; preserveAspect; startGroup] Equation (48) :[font = input; preserveAspect; startGroup] DSolve[ {x'[t] == 3 y[t] z[t], y'[t] == 3 x[t] z[t], z'[t] == - x[t] y[t]}, {x,y,z},t ] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] DSolve[{Derivative[1][x][t] == 3*y[t]*z[t], Derivative[1][y][t] == 3*x[t]*z[t], Derivative[1][z][t] == -(x[t]*y[t])}, {x, y, z}, t] ;[o] DSolve[{x'[t] == 3 y[t] z[t], y'[t] == 3 x[t] z[t], z'[t] == -(x[t] y[t])}, {x, y, z}, t] :[font = section; inactive; preserveAspect; startGroup] Equation (49) :[font = input; preserveAspect; startGroup] DSolve[ {x'[t] == a[t](y[t]^2-x[t]^2) + 2 b[t] x[t] y[t] + 2 c x[t], y'[t] == b[t](y[t]^2-x[t]^2) - 2 a[t] x[t] y[t] + 2 c y[t]}, {x,y},t ] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] DSolve[{Derivative[1][x][t] == 2*c*x[t] + 2*b[t]*x[t]*y[t] + a[t]*(-x[t]^2 + y[t]^2), Derivative[1][y][t] == 2*c*y[t] - 2*a[t]*x[t]*y[t] + b[t]*(-x[t]^2 + y[t]^2)}, {x, y}, t] ;[o] DSolve[{x'[t] == 2 c x[t] + 2 b[t] x[t] y[t] + 2 2 a[t] (-x[t] + y[t] ), y'[t] == 2 c y[t] - 2 a[t] x[t] y[t] + 2 2 b[t] (-x[t] + y[t] )}, {x, y}, t] :[font = section; inactive; preserveAspect; startGroup] Equation (50) :[font = input; preserveAspect; startGroup] DSolve[ {x'[t] == x[t] (1+Cos[t]/(2+Sin[t])), y'[t] == x[t] - y[t]}, {x,y},t ] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] DSolve[{Derivative[1][x][t] == (1 + Cos[t]/(2 + Sin[t]))*x[t], Derivative[1][y][t] == x[t] - y[t]}, {x, y}, t] ;[o] Cos[t] DSolve[{x'[t] == (1 + ----------) x[t], 2 + Sin[t] y'[t] == x[t] - y[t]}, {x, y}, t] :[font = section; inactive; preserveAspect; startGroup] Equation (51) :[font = input; preserveAspect; startGroup] DSolve[ {x'[t] == 9 x[t] + 2 y[t], y'[t] == x[t] + 8 y[t]}, {x,y}, t ] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] {{x -> Function[t, (E^(7*t)/3 + (2*E^(10*t))/3)*C[1] + ((-2*E^(7*t))/3 + (2*E^(10*t))/3)*C[2]], y -> Function[t, (-E^(7*t)/3 + E^(10*t)/3)*C[1] + ((2*E^(7*t))/3 + E^(10*t)/3)*C[2]]}} ;[o] 7 t 10 t E 2 E {{x -> Function[t, (---- + -------) C[1] + 3 3 7 t 10 t -2 E 2 E (------- + -------) C[2]], 3 3 7 t 10 t -E E y -> Function[t, (----- + -----) C[1] + 3 3 7 t 10 t 2 E E (------ + -----) C[2]]}} 3 3 :[font = section; inactive; preserveAspect; startGroup] Equation (52) :[font = input; preserveAspect; startGroup] DSolve[ {x'[t] - x[t] + 2 y[t] == 0, x''[t] - 2 y'[t] == 2 t - Cos[2 t]}, {x,y}, t ] :[font = message; inactive; preserveAspect] Solve::svars: Warning: Equations may not give solutions for all "solve" variables. :[font = message; inactive; preserveAspect; endGroup; endGroup] Solve::svars: Warning: Equations may not give solutions for all "solve" variables. :[font = section; inactive; preserveAspect; startGroup] Equation (53) :[font = input; preserveAspect; startGroup] DSolve[ {y1'[x] == -1/(x (x^2+1)) y1[x] + 1/(x^2 (x^2+1)) y2[x] + 1/x, y2'[x] == -x^2/(x^2+1) y1[x] + (2x^2+1)/(x (x^2+1)) y2[x] + 1}, {y1,y2}, x ] :[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup] DSolve[{Derivative[1][y1][x] == x^(-1) - y1[x]/(x*(1 + x^2)) + y2[x]/(x^2*(1 + x^2)), Derivative[1][y2][x] == 1 - (x^2*y1[x])/(1 + x^2) + ((1 + 2*x^2)*y2[x])/(x*(1 + x^2))}, {y1, y2}, x] ;[o] 1 y1[x] y2[x] DSolve[{y1'[x] == - - ---------- + -----------, x 2 2 2 x (1 + x ) x (1 + x ) 2 2 x y1[x] (1 + 2 x ) y2[x] y2'[x] == 1 - -------- + ----------------}, {y1, y2}, x] 2 2 1 + x x (1 + x ) ^*)