(*^ ::[ 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 = input; preserveAspect] << Calculus`DSolve` :[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 = message; inactive; preserveAspect] Solve::tdep: The equations appear to involve transcendental functions of the variables in an essentially non-algebraic way. :[font = message; inactive; preserveAspect] Solve::tdep: The equations appear to involve transcendental functions of the variables in an essentially non-algebraic way. :[font = message; inactive; preserveAspect] Solve::tdep: The equations appear to involve transcendental functions of the variables in an essentially non-algebraic way. :[font = message; inactive; preserveAspect] General::stop: Further output of Solve::tdep will be suppressed during this calculation. :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Solve[Log[y]^2/2 - #1/y == C[1], y] ;[o] 2 Log[y] #1 Solve[------- - -- == C[1], y] 2 y :[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 (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 = output; output; inactive; preserveAspect; endGroup; endGroup] $Aborted ;[o] $Aborted :[font = section; inactive; preserveAspect; startGroup] Equation (8) :[font = input; preserveAspect] ode = f[x y'[x]] == g[y'[x]] :[font = input; preserveAspect; endGroup] DSolve[ ode,y,x ] :[font = section; inactive; preserveAspect; startGroup] Equation (9) :[font = input; preserveAspect] ode = y'[x] == (3 x^2 - y[x]^2 - 7)/(Exp[y[x]]+2 x y[x]+1) :[font = input; preserveAspect; endGroup] DSolve[ ode,y,x ] :[font = section; inactive; preserveAspect; startGroup] Equation (10) :[font = input; preserveAspect] ode = y'[x] == (2 x^3 y[x] - y[x]^4) / (x^4 - 2 x y[x]^3) :[font = input; preserveAspect; endGroup] DSolve[ ode,y,x ] :[font = section; inactive; preserveAspect; startGroup] Equation (11) :[font = input; preserveAspect] ode = y'[x] (y'[x] + y[x]) == x (x + y[x]) :[font = input; preserveAspect] DSolve[ ode,y,x ] :[font = text; inactive; preserveAspect; plain; bold] Factored form of ode11: :[font = input; preserveAspect] odeFactored = (y'[x] + y[x] + x)(y'[x] - x) == 0 :[font = input; preserveAspect; endGroup] DSolve[ odeFactored,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 = message; inactive; preserveAspect] Solve::tdep: The equations appear to involve transcendental functions of the variables in an essentially non-algebraic way. :[font = message; inactive; preserveAspect] Solve::tdep: The equations appear to involve transcendental functions of the variables in an essentially non-algebraic way. :[font = message; inactive; preserveAspect] Solve::tdep: The equations appear to involve transcendental functions of the variables in an essentially non-algebraic way. :[font = message; inactive; preserveAspect] General::stop: Further output of Solve::tdep will be suppressed during this calculation. :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Solve[-y^(-1) - Log[y + #1^(2/3)]/(2*y^4) - Log[y^2 - y*#1^(2/3) + #1^(4/3)]/(2*y^4) + 1/(4*y^6*(y + #1^(2/3))) + 1/(2*y^5*(y^2 - y*#1^(2/3) + #1^(4/3))) - #1^(2/3)/(4*y^6*(y^2 - y*#1^(2/3) + #1^(4/3))) == C[1], y] ;[o] 2/3 2 2/3 4/3 1 Log[y + #1 ] Log[y - y #1 + #1 ] Solve[-(-) - -------------- - ------------------------- + y 4 4 2 y 2 y 1 1 ---------------- + --------------------------- - 6 2/3 5 2 2/3 4/3 4 y (y + #1 ) 2 y (y - y #1 + #1 ) 2/3 #1 --------------------------- == C[1], y] 6 2 2/3 4/3 4 y (y - y #1 + #1 ) :[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 = output; output; inactive; preserveAspect; endGroup; endGroup] $Aborted ;[o] $Aborted :[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] Solve::tdep: The equations appear to involve transcendental functions of the variables in an essentially non-algebraic way. :[font = message; inactive; preserveAspect] Solve::tdep: The equations appear to involve transcendental functions of the variables in an essentially non-algebraic way. :[font = message; inactive; preserveAspect] Solve::tdep: The equations appear to involve transcendental functions of the variables in an essentially non-algebraic way. :[font = message; inactive; preserveAspect] General::stop: Further output of Solve::tdep will be suppressed during this calculation. :[font = output; output; inactive; preserveAspect; endGroup; endGroup] {Solve[2*ArcTanh[(-4*(y - #1^2))^(1/2)/#1] - 4*ArcTanh[(-y + #1^2)^(1/2)/#1] + Log[y^2*(4*y - 3*#1^2)] == C[1], y], Solve[-2*ArcTanh[(-y + #1^2)^(1/2)/#1] + Log[(#1*(#1 + (-y + #1^2)^(1/2))^2)/y] == C[1], y]} ;[o] 2 Sqrt[-4 (y - #1 )] {Solve[2 ArcTanh[------------------] - #1 2 Sqrt[-y + #1 ] 2 2 4 ArcTanh[--------------] + Log[y (4 y - 3 #1 )] == #1 2 Sqrt[-y + #1 ] C[1], y], Solve[-2 ArcTanh[--------------] + #1 2 2 #1 (#1 + Sqrt[-y + #1 ]) Log[-------------------------] == C[1], y]} y :[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] {{y -> (Tan[#1 + C[1]]/E^#1 & )}} ;[o] Tan[#1 + C[1]] {{y -> (-------------- & )}} #1 E :[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] {{y -> ((-(E^(#1^2/2)*(2/Pi)^(1/2)*(#1^2)^(1/2)) + #1^2*C[1] + #1*(#1^2)^(1/2)*Erfi[#1/2^(1/2)])/ (#1*C[1] + (#1^2)^(1/2)*Erfi[#1/2^(1/2)]) & )}} ;[o] 2 #1 /2 2 2 2 {{y -> ((-(E Sqrt[--] Sqrt[#1 ]) + #1 C[1] + Pi 2 #1 #1 Sqrt[#1 ] Erfi[-------]) / Sqrt[2] 2 #1 (#1 C[1] + Sqrt[#1 ] Erfi[-------]) & )}} Sqrt[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 = output; output; inactive; preserveAspect; endGroup; endGroup] {{y -> (-(C[1]^(1/2)*(4*#1 + C[1])^(1/2))/2 & )}, {y -> ((C[1]^(1/2)*(4*#1 + C[1])^(1/2))/2 & )}, {y -> (-(C[1]^(1/2)*(4*#1 + 4*C[1])^(1/2)) & )}, {y -> (C[1]^(1/2)*(4*#1 + 4*C[1])^(1/2) & )}} ;[o] -(Sqrt[C[1]] Sqrt[4 #1 + C[1]]) {{y -> (------------------------------- & )}, 2 Sqrt[C[1]] Sqrt[4 #1 + C[1]] {y -> (---------------------------- & )}, 2 {y -> (-(Sqrt[C[1]] Sqrt[4 #1 + 4 C[1]]) & )}, {y -> (Sqrt[C[1]] Sqrt[4 #1 + 4 C[1]] & )}} :[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] Solve::tdep: The equations appear to involve transcendental functions of the variables in an essentially non-algebraic way. :[font = message; inactive; preserveAspect] Solve::tdep: The equations appear to involve transcendental functions of the variables in an essentially non-algebraic way. :[font = message; inactive; preserveAspect] Solve::tdep: The equations appear to involve transcendental functions of the variables in an essentially non-algebraic way. :[font = message; inactive; preserveAspect] General::stop: Further output of Solve::tdep will be suppressed during this calculation. :[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup] {Solve[Log[y + (y^2 - 4*#1^2)^(1/2)] + y^2/(4*#1^2) - (y*(y^2 - 4*#1^2)^(1/2))/(4*#1^2) == C[1], y], Solve[Log[#1^2/(y + (y^2 - 4*#1^2)^(1/2))] + y^2/(4*#1^2) + (y*(y^2 - 4*#1^2)^(1/2))/(4*#1^2) == C[1] , y]} ;[o] 2 2 2 y {Solve[Log[y + Sqrt[y - 4 #1 ]] + ----- - 2 4 #1 2 2 y Sqrt[y - 4 #1 ] ------------------ == C[1], y], 2 4 #1 2 2 #1 y Solve[Log[--------------------] + ----- + 2 2 2 y + Sqrt[y - 4 #1 ] 4 #1 2 2 y Sqrt[y - 4 #1 ] ------------------ == C[1], y]} 2 4 #1 :[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 (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 = output; output; inactive; preserveAspect; endGroup; endGroup] {{y -> ((-1 + (1 - 4*C[2])^(1/2)* Tanh[((-#1 - C[1])*(1 - 4*C[2])^(1/2))/2])/2 & )}} ;[o] {{y -> ((-1 + Sqrt[1 - 4 C[2]] (-#1 - C[1]) Sqrt[1 - 4 C[2]] Tanh[-----------------------------]) / 2 & )}} 2 :[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 = output; output; inactive; preserveAspect; endGroup; endGroup] {-(3^(1/2)*Integrate[(2/y + 6*y^2*C[1] + 6*y^2*Log[y])^ (-1/2), y]) == C[2] + #1, 3^(1/2)*Integrate[(2/y + 6*y^2*C[1] + 6*y^2*Log[y])^ (-1/2), y] == C[2] + #1} ;[o] 1 {-(Sqrt[3] Integrate[---------------------------------, 2 2 2 Sqrt[- + 6 y C[1] + 6 y Log[y]] y y]) == C[2] + #1, 1 Sqrt[3] Integrate[---------------------------------, y] == 2 2 2 Sqrt[- + 6 y C[1] + 6 y Log[y]] y C[2] + #1} :[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] {{y -> (C[2] + (Pi^(1/2)*C[1]*Erf[#1])/2 + #1 & )}} ;[o] Sqrt[Pi] C[1] Erf[#1] {{y -> (C[2] + --------------------- + #1 & )}} 2 :[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 = message; inactive; preserveAspect] DSolve::dnim: Built-in procedures cannot solve this differential equation. :[font = message; inactive; preserveAspect] DSolve::dnim: Built-in procedures cannot solve this differential equation. :[font = message; inactive; preserveAspect] General::stop: Further output of DSolve::dnim will be suppressed during this calculation. :[font = output; output; inactive; preserveAspect; endGroup; endGroup] $Aborted ;[o] $Aborted :[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 = output; output; inactive; preserveAspect; endGroup; endGroup] {{y -> ((-1 + (1 - 4*C[2])^(1/2)* Tanh[((1 - 4*C[2])^(1/2)*(-C[1] - Log[#1]))/2])/2 \ & )}} ;[o] {{y -> ((-1 + Sqrt[1 - 4 C[2]] Sqrt[1 - 4 C[2]] (-C[1] - Log[#1]) Tanh[----------------------------------]) / 2 & )}} 2 :[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 = output; output; inactive; preserveAspect; endGroup; endGroup] {{y -> (E^(#1^2/2 + #1^3/6 + #1*(-1 + C[1]) - Log[1 - #1] + #1*Log[1 - #1])*C[2] & )}} ;[o] 2 3 {{y -> (Power[E, #1 /2 + #1 /6 + #1 (-1 + C[1]) - Log[1 - #1] + #1 Log[1 - #1]] C[2] & )}} :[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 = output; output; inactive; preserveAspect; endGroup; endGroup] {{y -> (-(2*C[1] + 2*C[2]*Log[#1])^(1/2) & )}, {y -> ((2*C[1] + 2*C[2]*Log[#1])^(1/2) & )}} ;[o] {{y -> (-Sqrt[2 C[1] + 2 C[2] Log[#1]] & )}, {y -> (Sqrt[2 C[1] + 2 C[2] Log[#1]] & )}} :[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 ;[o] changed :[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] {{y -> ((2*C[1]*Gamma[2/3]* LaguerreL[2/3, -2/3, (4*#1^(3/2))/3])/ (3*E^((4*#1^(3/2))/3)) + (2*(2/9)^(1/3)*C[2]*(#1^(3/2))^(2/3))/ E^((4*#1^(3/2))/3) & )}} ;[o] 3/2 2 2 2 4 #1 2 C[1] Gamma[-] LaguerreL[-, -(-), -------] 3 3 3 3 {{y -> (------------------------------------------- + 3/2 (4 #1 )/3 3 E 2 1/3 3/2 2/3 2 (-) C[2] (#1 ) 9 ------------------------ & )}} 3/2 (4 #1 )/3 E :[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] General::intinit: Loading integration packages -- please wait. :[font = message; inactive; preserveAspect] 1 Power::infy: Infinite expression ------- encountered. Sqrt[0] :[font = message; inactive; preserveAspect] Infinity::indet: Indeterminate expression 0 ComplexInfinity encountered. :[font = message; inactive; preserveAspect] 1 Power::infy: Infinite expression ------- encountered. Sqrt[0] :[font = message; inactive; preserveAspect] Infinity::indet: Indeterminate expression 0 ComplexInfinity encountered. :[font = output; output; inactive; preserveAspect; endGroup; endGroup] {{}} ;[o] {{}} :[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; 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 = 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 (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 = 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 (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 + ComplexInfinity 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 -> ((BesselK[0, I*2^(1/2)*#1]*C[1])/Pi^(1/2) + BesselI[0, I*2^(1/2)*#1]* (1 + C[1]*DirectedInfinity[-1]) & )}} ;[o] BesselK[0, I Sqrt[2] #1] C[1] {{y -> (----------------------------- + Sqrt[Pi] BesselI[0, I Sqrt[2] #1] (1 + C[1] (-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] Solve::ifun: Warning: Inverse functions are being used by Solve, so some solutions may not be found. :[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 -> ((E^(2*#1^(1/2))*(1 + E^(-4*#1^(1/2))))/2 & )}, {y -> ((1 + E^(4*#1^(1/2)))/(2*E^(2*#1^(1/2))) & )}} ;[o] 2 Sqrt[#1] -4 Sqrt[#1] E (1 + E ) {{y -> (------------------------------ & )}, 2 4 Sqrt[#1] 1 + E {y -> (--------------- & )}} 2 Sqrt[#1] 2 E :[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 = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup; endGroup] {{y -> ((-3*2^(1/3))/ (162*#1 + (2916 + 26244*#1^2)^(1/2))^(1/3) + (162*#1 + (2916 + 26244*#1^2)^(1/2))^(1/3)/ (3*2^(1/3)) & )}, {y -> ((3*(1 + I*3^(1/2)))/ (2^(2/3)*(162*#1 + (2916 + 26244*#1^2)^(1/2))^(1/3)) - ((1 - I*3^(1/2))* (162*#1 + (2916 + 26244*#1^2)^(1/2))^(1/3))/ (6*2^(1/3)) & )}, {y -> ((3*(1 - I*3^(1/2)))/ (2^(2/3)*(162*#1 + (2916 + 26244*#1^2)^(1/2))^(1/3)) - ((1 + I*3^(1/2))* (162*#1 + (2916 + 26244*#1^2)^(1/2))^(1/3))/ (6*2^(1/3)) & )}} ;[o] 1/3 -3 2 {{y -> (------------------------------------ + 2 1/3 (162 #1 + Sqrt[2916 + 26244 #1 ]) 2 1/3 (162 #1 + Sqrt[2916 + 26244 #1 ]) ------------------------------------ & )}, 1/3 3 2 3 (1 + I Sqrt[3]) {y -> (----------------------------------------- - 2/3 2 1/3 2 (162 #1 + Sqrt[2916 + 26244 #1 ]) 2 1/3 (1 - I Sqrt[3]) (162 #1 + Sqrt[2916 + 26244 #1 ]) ----------------------------------------------------\ 1/3 6 2 & )}, {y -> 3 (1 - I Sqrt[3]) (----------------------------------------- - 2/3 2 1/3 2 (162 #1 + Sqrt[2916 + 26244 #1 ]) 2 1/3 (1 + I Sqrt[3]) (162 #1 + Sqrt[2916 + 26244 #1 ]) ----------------------------------------------------\ 1/3 6 2 & )}} :[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] {{Integrate[1/ ((-Calculus`DSolve`Private`v$572^2 - 2*C[1])^(1/2)* (Calculus`DSolve`Private`v$572^2 + 2*C[2])^(1/2)), {Calculus`DSolve`Private`v$572, 0, x}]/3^(1/2) == C[3] + #1, y -> (x^2 + 2*C[2])^(1/2), z -> (-x^2 - 2*C[1])^(1/2)/3^(1/2)}, {-(Integrate[1/ ((-Calculus`DSolve`Private`v$587^2 - 2*C[1])^(1/2)* (Calculus`DSolve`Private`v$587^2 + 2*C[2])^(1/2))\ , {Calculus`DSolve`Private`v$587, 0, x}]/3^(1/2)) =\ = C[3] + #1, y -> (x^2 + 2*C[2])^(1/2), z -> -((-x^2 - 2*C[1])^(1/2)/3^(1/2))}, {-(Integrate[1/ ((-Calculus`DSolve`Private`v$602^2 - 2*C[1])^(1/2)* (Calculus`DSolve`Private`v$602^2 + 2*C[2])^(1/2))\ , {Calculus`DSolve`Private`v$602, 0, x}]/3^(1/2)) =\ = C[3] + #1, y -> -(x^2 + 2*C[2])^(1/2), z -> (-x^2 - 2*C[1])^(1/2)/3^(1/2)}, {Integrate[1/ ((-Calculus`DSolve`Private`v$617^2 - 2*C[1])^(1/2)* (Calculus`DSolve`Private`v$617^2 + 2*C[2])^(1/2)), {Calculus`DSolve`Private`v$617, 0, x}]/3^(1/2) == C[3] + #1, y -> -(x^2 + 2*C[2])^(1/2), z -> -((-x^2 - 2*C[1])^(1/2)/3^(1/2))}} ;[o] {{Integrate[1 / 2 (Sqrt[-Calculus`DSolve`Private`v$572 - 2 C[1]] 2 Sqrt[Calculus`DSolve`Private`v$572 + 2 C[2]]), {Calculus`DSolve`Private`v$572, 0, x}] / Sqrt[3] == 2 C[3] + #1, y -> Sqrt[x + 2 C[2]], 2 Sqrt[-x - 2 C[1]] z -> ------------------}, Sqrt[3] {-(Integrate[1 / 2 (Sqrt[-Calculus`DSolve`Private`v$587 - 2 C[1]] 2 Sqrt[Calculus`DSolve`Private`v$587 + 2 C[2]]), {Calculus`DSolve`Private`v$587, 0, x}] / Sqrt[3]) == 2 C[3] + #1, y -> Sqrt[x + 2 C[2]], 2 Sqrt[-x - 2 C[1]] z -> -(------------------)}, Sqrt[3] {-(Integrate[1 / 2 (Sqrt[-Calculus`DSolve`Private`v$602 - 2 C[1]] 2 Sqrt[Calculus`DSolve`Private`v$602 + 2 C[2]]), {Calculus`DSolve`Private`v$602, 0, x}] / Sqrt[3]) == 2 C[3] + #1, y -> -Sqrt[x + 2 C[2]], 2 Sqrt[-x - 2 C[1]] z -> ------------------}, Sqrt[3] {Integrate[1 / 2 (Sqrt[-Calculus`DSolve`Private`v$617 - 2 C[1]] 2 Sqrt[Calculus`DSolve`Private`v$617 + 2 C[2]]), {Calculus`DSolve`Private`v$617, 0, x}] / Sqrt[3] == 2 C[3] + #1, y -> -Sqrt[x + 2 C[2]], 2 Sqrt[-x - 2 C[1]] z -> -(------------------)}} Sqrt[3] :[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] {{x -> E^(Log[2 + Sin[#1]] + #1)*C[2], y -> C[1]/E^#1 + (E^#1* (5*C[2] - C[2]*Cos[#1] + 2*C[2]*Sin[#1]))/5}} ;[o] Log[2 + Sin[#1]] + #1 {{x -> E C[2], #1 C[1] E (5 C[2] - C[2] Cos[#1] + 2 C[2] Sin[#1]) y -> ---- + --------------------------------------------}} #1 5 E :[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 ) ^*)