Single equations without initial conditions

First order equations

<< Calculus`DSolve`

Equation (3)

ode = y'[x] == y[x]/(y[x] Log[y[x]]+x)

Derivative[1][y][x] == y[x]/(x + Log[y[x]]*y[x])

DSolve[ ode,y,x ]

Solve::tdep: 
   The equations appear to involve transcendental functions
     of the variables in an essentially non-algebraic way.

Solve::tdep: 
   The equations appear to involve transcendental functions
     of the variables in an essentially non-algebraic way.

Solve::tdep: 
   The equations appear to involve transcendental functions
     of the variables in an essentially non-algebraic way.

General::stop: 
   Further output of Solve::tdep
     will be suppressed during this calculation.

Solve[Log[y]^2/2 - #1/y == C[1], y]

Equation (4)

ode = 2 y[x] y'[x]^2 - 2 x y'[x] - y[x] == 0

-y[x] - 2*x*Derivative[1][y][x] + 
 
   2*y[x]*Derivative[1][y][x]^2 == 0

DSolve[ ode,y,x ]

DSolve::dnim: 
   Built-in procedures cannot solve this differential
     equation.

DSolve[-y[x] - 2*x*Derivative[1][y][x] + 
 
    2*y[x]*Derivative[1][y][x]^2 == 0, y, x]

Equation (7)

ode= (x^2-1) y'[x]^2 - 2 x y[x]y'[x] + y^2 - 1 == 0

-1 + y^2 - 2*x*y[x]*Derivative[1][y][x] + 
 
   (-1 + x^2)*Derivative[1][y][x]^2 == 0

DSolve[ ode,y,x ]

$Aborted

Equation (8)

ode = f[x y'[x]] == g[y'[x]]

DSolve[ ode,y,x ]

Equation (9)

ode = y'[x] == (3 x^2 - y[x]^2 - 7)/(Exp[y[x]]+2 x y[x]+1)

DSolve[ ode,y,x ]

Equation (10)

ode = y'[x] == (2 x^3 y[x] - y[x]^4) / (x^4 - 2 x y[x]^3)

DSolve[ ode,y,x ]

Equation (11)

ode = y'[x] (y'[x] + y[x]) == x (x + y[x])

DSolve[ ode,y,x ]

Factored form of ode11:

odeFactored = (y'[x] + y[x] + x)(y'[x] - x) == 0


DSolve[ odeFactored,y,x ]

Equation (12)

ode = y'[x] == x/(x^2 y[x]^2 + y[x]^5)

Derivative[1][y][x] == x/(x^2*y[x]^2 + y[x]^5)

DSolve[ ode,y,x ]

Solve::tdep: 
   The equations appear to involve transcendental functions
     of the variables in an essentially non-algebraic way.

Solve::tdep: 
   The equations appear to involve transcendental functions
     of the variables in an essentially non-algebraic way.

Solve::tdep: 
   The equations appear to involve transcendental functions
     of the variables in an essentially non-algebraic way.

General::stop: 
   Further output of Solve::tdep
     will be suppressed during this calculation.

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]

Equation (13)

ode = y[x] == 2 x y'[x] - a y'[x]^3

y[x] == 2*x*Derivative[1][y][x] - a*Derivative[1][y][x]^3

DSolve[ ode,y,x ]

$Aborted

Equation (14)

ode = y[x] == 2 x y'[x] - y'[x]^2

y[x] == 2*x*Derivative[1][y][x] - Derivative[1][y][x]^2

DSolve[ ode,y,x ]

Solve::tdep: 
   The equations appear to involve transcendental functions
     of the variables in an essentially non-algebraic way.

Solve::tdep: 
   The equations appear to involve transcendental functions
     of the variables in an essentially non-algebraic way.

Solve::tdep: 
   The equations appear to involve transcendental functions
     of the variables in an essentially non-algebraic way.

General::stop: 
   Further output of Solve::tdep
     will be suppressed during this calculation.

{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]}

Equation (15)

ode = y'[x] == Exp[x] y[x]^2 - y[x] + Exp[-x]

Derivative[1][y][x] == E^(-x) - y[x] + E^x*y[x]^2

DSolve[ ode,y,x ]

{{y -> (Tan[#1 + C[1]]/E^#1 & )}}

Equation (16)

ode = y'[x] == y[x]^2 - x y[x] + 1

Derivative[1][y][x] == 1 - x*y[x] + y[x]^2

DSolve[ ode,y,x ]

{{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)]) & )}}

Equation (18)

ode = y[x] == 2 x y'[x] + y[x] y'[x]^2

y[x] == 2*x*Derivative[1][y][x] + y[x]*Derivative[1][y][x]^2

DSolve[ ode,y,x ]

{{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) & )}}

Equation (19)

ode = y[x] y'[x] - x y'[x]^2 == x

y[x]*Derivative[1][y][x] - x*Derivative[1][y][x]^2 == x

DSolve[ ode,y,x ]

Solve::tdep: 
   The equations appear to involve transcendental functions
     of the variables in an essentially non-algebraic way.

Solve::tdep: 
   The equations appear to involve transcendental functions
     of the variables in an essentially non-algebraic way.

Solve::tdep: 
   The equations appear to involve transcendental functions
     of the variables in an essentially non-algebraic way.

General::stop: 
   Further output of Solve::tdep
     will be suppressed during this calculation.

{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]}

Second order equations

Equation (21)

ode = (x^2-x) u''[x] + (2 x^2+4 x -3) u'[x] +  8 x u[x] == 1

8*x*u[x] + (-3 + 4*x + 2*x^2)*Derivative[1][u][x] + 
 
   (-x + x^2)*Derivative[2][u][x] == 1

DSolve[ ode,u,x ]

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]

Equation (22)

ode = (x^2-x) w''[x] + (1-2 x^2) w'[x] + (4 x-2) w[x] == 0

(-2 + 4*x)*w[x] + (1 - 2*x^2)*Derivative[1][w][x] + 
 
   (-x + x^2)*Derivative[2][w][x] == 0

DSolve[ ode,w,x ]

DSolve[(-2 + 4*x)*w[x] + (1 - 2*x^2)*Derivative[1][w][x] + 
 
    (-x + x^2)*Derivative[2][w][x] == 0, w, x]

Equation (23)

ode = y''[x] - y'[x] == 2 y[x] y'[x]

-Derivative[1][y][x] + Derivative[2][y][x] == 
 
  2*y[x]*Derivative[1][y][x]

DSolve[ ode,y,x ]

{{y -> ((-1 + (1 - 4*C[2])^(1/2)*
 
          Tanh[((-#1 - C[1])*(1 - 4*C[2])^(1/2))/2])/2 & )}}

Equation (24)

ode = y''[x]/y[x] - y'[x]^2/y[x]^2 - 1 + 1/y[x]^3 == 0

-1 + y[x]^(-3) - Derivative[1][y][x]^2/y[x]^2 + 
 
   Derivative[2][y][x]/y[x] == 0

DSolve[ ode,y,x ]

{-(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}

Equation (25)

ode = y''[x] + 2 x y'[x] == 2 x

2*x*Derivative[1][y][x] + Derivative[2][y][x] == 2*x

DSolve[ ode,y,x ]

{{y -> (C[2] + (Pi^(1/2)*C[1]*Erf[#1])/2 + #1 & )}}

Equation (26)

ode = 2 y[x] y''[x] - y'[x]^2 == 1/3 (y'[x] - x y''[x])^2

-Derivative[1][y][x]^2 + 2*y[x]*Derivative[2][y][x] == 
 
  (Derivative[1][y][x] - x*Derivative[2][y][x])^2/3

DSolve[ ode,y,x ]

DSolve::dnim: 
   Built-in procedures cannot solve this differential
     equation.

DSolve::dnim: 
   Built-in procedures cannot solve this differential
     equation.

DSolve::dnim: 
   Built-in procedures cannot solve this differential
     equation.

General::stop: 
   Further output of DSolve::dnim
     will be suppressed during this calculation.

$Aborted

Equation (27)

ode = x y''[x] == 2 y[x] y'[x]

x*Derivative[2][y][x] == 2*y[x]*Derivative[1][y][x]

DSolve[ ode,y,x ]

{{y -> ((-1 + (1 - 4*C[2])^(1/2)*
 
          Tanh[((1 - 4*C[2])^(1/2)*(-C[1] - Log[#1]))/2])/2 \
 
       & )}}

Equation (28)

ode = (1-x)(y[x] y''[x] - y'[x]^2) + x^2 y[x]^2 == 0

x^2*y[x]^2 + (1 - x)*(-Derivative[1][y][x]^2 + 
 
      y[x]*Derivative[2][y][x]) == 0

DSolve[ ode,y,x ]

{{y -> (E^(#1^2/2 + #1^3/6 + #1*(-1 + C[1]) - Log[1 - #1] + 
 
          #1*Log[1 - #1])*C[2] & )}}

Equation (29)

ode = x y[x] y''[x] + x y'[x]^2 + y[x] y'[x] == 0

y[x]*Derivative[1][y][x] + x*Derivative[1][y][x]^2 + 
 
   x*y[x]*Derivative[2][y][x] == 0

DSolve[ ode,y,x ]

{{y -> (-(2*C[1] + 2*C[2]*Log[#1])^(1/2) & )}, 
 
  {y -> ((2*C[1] + 2*C[2]*Log[#1])^(1/2) & )}}

Equation (30)

ode = y''[x]^2 - 2 y'[x] y''[x] + 2 y[x] y'[x] - y[x]^2 == 0

     2                                         2
-y[x]  + 2 y[x] y'[x] - 2 y'[x] y''[x] + y''[x]  == 0

DSolve[ ode,y,x ]


                C[1]    #1                    #1         #1
Out[3]= {{y -> (---- + E   C[2] & )}, {y -> (E   C[1] + E   #1 C[2] & )}}
                 #1
                E

Equation (32)

ode = y''[x] - 2 x y'[x] + 2 y[x] == 3

2*y[x] - 2*x*Derivative[1][y][x] + Derivative[2][y][x] == 3

DSolve[ ode,y,x ]

DSolve[2*y[x] - 2*x*Derivative[1][y][x] + 
 
    Derivative[2][y][x] == 3, y, x]

Equation (33)

ode = Sqrt[x] y''[x] + 2 x y'[x] + 3 y[x] == 0

3*y[x] + 2*x*Derivative[1][y][x] + 
 
   x^(1/2)*Derivative[2][y][x] == 0

DSolve[ ode,y,x ]

{{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) & )}}

Equation (34)

ode = x^2 y''[x] + 3 x y'[x] + 2 y[x] == 1/(y[x]^3 x^4)

DSolve[ ode,y,x ]

General::intinit: 
   Loading integration packages -- please wait.

                                    1
Power::infy: Infinite expression ------- encountered.
                                 Sqrt[0]

Infinity::indet: 
   Indeterminate expression 0 ComplexInfinity encountered.

                                    1
Power::infy: Infinite expression ------- encountered.
                                 Sqrt[0]

Infinity::indet: 
   Indeterminate expression 0 ComplexInfinity encountered.

{{}}

Equation (35)

ode = y''[x] - 2/x^2 y[x] == 7 x^4 + 3 x^3

(-2*y[x])/x^2 + Derivative[2][y][x] == 3*x^3 + 7*x^4

DSolve[ ode,y,x ]

DSolve[(-2*y[x])/x^2 + Derivative[2][y][x] == 3*x^3 + 7*x^4, 
 
  y, x]

Higher order equations

Equation (38)

ode = y''''[x] - 4/x^2 y''[x] + 8/x^3 y'[x] - 8/x^4 y[x] == 0

(-8*y[x])/x^4 + (8*Derivative[1][y][x])/x^3 - 
 
   (4*Derivative[2][y][x])/x^2 + Derivative[4][y][x] == 0

DSolve[ ode,y,x ]

DSolve::dnim: 
   Built-in procedures cannot solve this differential
     equation.

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]

Equation (39)

ode = (1+x+x^2) y'''[x] + (3 + 6 x) y''[x] + 6 y'[x] == 6 x

6*Derivative[1][y][x] + (3 + 6*x)*Derivative[2][y][x] + 
 
   (1 + x + x^2)*Derivative[3][y][x] == 6*x

DSolve[ ode,y,x ]

DSolve::dnim: 
   Built-in procedures cannot solve this differential
     equation.

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]

Equation (40)

ode = (y'[x]^2+1) y'''[x] - 3 y'[x] y''[x]^2 == 0

-3*Derivative[1][y][x]*Derivative[2][y][x]^2 + 
 
   (1 + Derivative[1][y][x]^2)*Derivative[3][y][x] == 0

DSolve[ ode,y,x ]

DSolve::dnim: 
   Built-in procedures cannot solve this differential
     equation.

DSolve::dnim: 
   Built-in procedures cannot solve this differential
     equation.

DSolve[-3*Derivative[1][y][x]*Derivative[2][y][x]^2 + 
 
    (1 + Derivative[1][y][x]^2)*Derivative[3][y][x] == 0, y, 
 
  x]

Equation (41)

ode = 3 y''[x] y''''[x] - 5 y'''[x]^2 == 0

-5*Derivative[3][y][x]^2 + 
 
   3*Derivative[2][y][x]*Derivative[4][y][x] == 0

DSolve[ ode,y,x ]

DSolve::dnim: 
   Built-in procedures cannot solve this differential
     equation.

DSolve[-5*Derivative[3][y][x]^2 + 
 
    3*Derivative[2][y][x]*Derivative[4][y][x] == 0, y, x]

Special equations

Equation (42)

ode = y'[t] + a y[t-1] == 0

a*y[-1 + t] + Derivative[1][y][t] == 0

DSolve[ ode,y,t ]

{{y -> Function[t, C[1] - 
 
      a*Integrate[y[-1 + DSolve`t], {DSolve`t, 0, t}]]}}

Equation (43)

ode = D[y[x,a],x] == a y[x,a]

Derivative[1, 0][y][x, a] == a*y[x, a]

DSolve[ ode,y,x ]

DSolve::deqx: 
   Supplied equations are not differential equations of the
     given functions.

DSolve[Derivative[1, 0][y][x, a] == a*y[x, a], y, x]

Single equations with initial conditions

Equation (45)

ode = x y''[x] + y'[x] + 2 x y[x] == 0

2*x*y[x] + Derivative[1][y][x] + x*Derivative[2][y][x] == 0

DSolve[ {ode, y[0]==1,y'[0]==0},y,x ]

Infinity::indet: 
   Indeterminate expression 
    ComplexInfinity + ComplexInfinity encountered.

Solve::svars: 
   Warning: Equations may not give solutions for all "solve"
     variables.

{{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]) & )}}

Equation (46)

ode = x y'[x]^2 - y[x]^2 + 1 == 0

1 - y[x]^2 + x*Derivative[1][y][x]^2 == 0

DSolve[ {ode, y[0]==1},y,x ]

Solve::ifun: 
   Warning: Inverse functions are being used by Solve, so
     some solutions may not be found.

Solve::ifun: 
   Warning: Inverse functions are being used by Solve, so
     some solutions may not be found.

{{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))) & )}}

Equation (47)

ode = y''[x] + y[x] y'[x]^3 == 0

y[x]*Derivative[1][y][x]^3 + Derivative[2][y][x] == 0

DSolve[ {ode, y[0]==0, y'[0]==2},y,x ]

{{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)) & )}}

Systems of equations

Equation (48)

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 ]

{{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))}}

Equation (49)

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 ]

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]

Equation (50)

DSolve[
  {x'[t] == x[t] (1+Cos[t]/(2+Sin[t])),
   y'[t] == x[t] - y[t]},
{x,y},t ]

{{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}}

Equation (52)

DSolve[ {x'[t] - x[t] + 2 y[t] == 0,
         x''[t] - 2 y'[t] == 2 t - Cos[2 t]},
   {x,y}, t ]


Solve::svars: 
   Warning: Equations may not give solutions for all "solve"
     variables.

Solve::svars: 
   Warning: Equations may not give solutions for all "solve"
     variables.

Equation (53)

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 ]


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]

a0=104/25*x^10+(274/25-22/15*Sqrt[-222])*x^8+(7754/75-68/15*Sqrt[-222])*x^6+
(11248/75-194/15*Sqrt[-222])*x^4+(29452/75-296/5*Sqrt[-222])*x^2-
10952/5-148/3*Sqrt[-222]
a2=x^12+2*x^10+151/3*x^8+296/3*x^6+5920/9*x^4+10952/9*x^2+5476/9
ode=a2*y''[x]-a0*y[x]==0
In[8]:= DSolve[ode,y[x],x]

                    10952               74     29452   296 I             2
Out[8]= DSolve[-((-(-----) - 148 I Sqrt[--] + (----- - ----- Sqrt[222]) x  + 
                      5                 3       75       5
 
             11248   194 I      74    4    7754   68 I      74    6
>           (----- - ----- Sqrt[--]) x  + (---- - ---- Sqrt[--]) x  + 
              75       5        3           75     5        3
 
                                            10
             274   22 I      74    8   104 x
>           (--- - ---- Sqrt[--]) x  + -------) y[x]) + 
             25     5        3           25
 
                      2         4        6        8
        5476   10952 x    5920 x    296 x    151 x       10    12
>      (---- + -------- + ------- + ------ + ------ + 2 x   + x  ) y''[x] == 0
         9        9          9        3        3
 
>     , y[x], x]