4.34.35 \(x \left (\text {a0}+x^2\right ) y''(x)+\left (\text {a1}+\text {b1} x^2\right ) y'(x)+\text {a2} x y(x)=0\)

ODE
\[ x \left (\text {a0}+x^2\right ) y''(x)+\left (\text {a1}+\text {b1} x^2\right ) y'(x)+\text {a2} x y(x)=0 \] ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.570333 (sec), leaf count = 177

\[\left \{\left \{y(x)\to c_2 \text {a0}^{\frac {1}{2} \left (\frac {\text {a1}}{\text {a0}}-1\right )} x^{1-\frac {\text {a1}}{\text {a0}}} \, _2F_1\left (\frac {\text {a0} \left (\text {b1}-\sqrt {(\text {b1}-1)^2-4 \text {a2}}+1\right )-2 \text {a1}}{4 \text {a0}},\frac {\text {a0} \left (\text {b1}+\sqrt {(\text {b1}-1)^2-4 \text {a2}}+1\right )-2 \text {a1}}{4 \text {a0}};\frac {3}{2}-\frac {\text {a1}}{2 \text {a0}};-\frac {x^2}{\text {a0}}\right )+c_1 \, _2F_1\left (\frac {1}{4} \left (\text {b1}-\sqrt {(\text {b1}-1)^2-4 \text {a2}}-1\right ),\frac {1}{4} \left (\text {b1}+\sqrt {(\text {b1}-1)^2-4 \text {a2}}-1\right );\frac {\text {a0}+\text {a1}}{2 \text {a0}};-\frac {x^2}{\text {a0}}\right )\right \}\right \}\]

Maple ✓
cpu = 0.685 (sec), leaf count = 203

\[\left [y \left (x \right ) = \textit {\_C1} \,x^{\frac {\mathit {a0} -\mathit {a1}}{\mathit {a0}}} \left (x^{2}+\mathit {a0} \right )^{\frac {\left (-\mathit {b1} +2\right ) \mathit {a0} +\mathit {a1}}{2 \mathit {a0}}} \hypergeom \left (\left [-\frac {\sqrt {\mathit {b1}^{2}-4 \mathit {a2} -2 \mathit {b1} +1}}{4}-\frac {\mathit {b1}}{4}+\frac {5}{4}, \frac {\sqrt {\mathit {b1}^{2}-4 \mathit {a2} -2 \mathit {b1} +1}}{4}-\frac {\mathit {b1}}{4}+\frac {5}{4}\right ], \left [\frac {3 \mathit {a0} -\mathit {a1}}{2 \mathit {a0}}\right ], -\frac {x^{2}}{\mathit {a0}}\right )+\textit {\_C2} \left (x^{2}+\mathit {a0} \right )^{\frac {\left (-\mathit {b1} +2\right ) \mathit {a0} +\mathit {a1}}{2 \mathit {a0}}} \hypergeom \left (\left [-\frac {\sqrt {\mathit {b1}^{2}-4 \mathit {a2} -2 \mathit {b1} +1}\, \mathit {a0} +\mathit {a0} \mathit {b1} -3 \mathit {a0} -2 \mathit {a1}}{4 \mathit {a0}}, \frac {\sqrt {\mathit {b1}^{2}-4 \mathit {a2} -2 \mathit {b1} +1}\, \mathit {a0} +\left (3-\mathit {b1} \right ) \mathit {a0} +2 \mathit {a1}}{4 \mathit {a0}}\right ], \left [\frac {\mathit {a0} +\mathit {a1}}{2 \mathit {a0}}\right ], -\frac {x^{2}}{\mathit {a0}}\right )\right ]\] Mathematica raw input

DSolve[a2*x*y[x] + (a1 + b1*x^2)*y'[x] + x*(a0 + x^2)*y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> C[1]*Hypergeometric2F1[(-1 - Sqrt[-4*a2 + (-1 + b1)^2] + b1)/4, (-1 + 
Sqrt[-4*a2 + (-1 + b1)^2] + b1)/4, (a0 + a1)/(2*a0), -(x^2/a0)] + a0^((-1 + a1/a
0)/2)*x^(1 - a1/a0)*C[2]*Hypergeometric2F1[(-2*a1 + a0*(1 - Sqrt[-4*a2 + (-1 + b
1)^2] + b1))/(4*a0), (-2*a1 + a0*(1 + Sqrt[-4*a2 + (-1 + b1)^2] + b1))/(4*a0), 3
/2 - a1/(2*a0), -(x^2/a0)]}}

Maple raw input

dsolve(x*(x^2+a0)*diff(diff(y(x),x),x)+(b1*x^2+a1)*diff(y(x),x)+a2*x*y(x) = 0, y(x))

Maple raw output

[y(x) = _C1*x^((a0-a1)/a0)*(x^2+a0)^(1/2*((-b1+2)*a0+a1)/a0)*hypergeom([-1/4*(b1
^2-4*a2-2*b1+1)^(1/2)-1/4*b1+5/4, 1/4*(b1^2-4*a2-2*b1+1)^(1/2)-1/4*b1+5/4],[1/2*
(3*a0-a1)/a0],-1/a0*x^2)+_C2*(x^2+a0)^(1/2*((-b1+2)*a0+a1)/a0)*hypergeom([-1/4/a
0*((b1^2-4*a2-2*b1+1)^(1/2)*a0+a0*b1-3*a0-2*a1), 1/4*((b1^2-4*a2-2*b1+1)^(1/2)*a
0+(3-b1)*a0+2*a1)/a0],[1/2/a0*(a0+a1)],-1/a0*x^2)]