4.30.37 \(\text {a1} x y'(x)+y(x) \left (\text {a2}+\text {b2} x^2\right )+x^2 y''(x)=0\)

ODE
\[ \text {a1} x y'(x)+y(x) \left (\text {a2}+\text {b2} x^2\right )+x^2 y''(x)=0 \] ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.0199611 (sec), leaf count = 78

\[\left \{\left \{y(x)\to x^{\frac {1}{2}-\frac {\text {a1}}{2}} \left (c_1 J_{\frac {1}{2} \sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}}\left (\sqrt {\text {b2}} x\right )+c_2 Y_{\frac {1}{2} \sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}}\left (\sqrt {\text {b2}} x\right )\right )\right \}\right \}\]

Maple ✓
cpu = 0.042 (sec), leaf count = 59

\[ \left \{ y \left ( x \right ) ={x}^{-{\frac {{\it a1}}{2}}+{\frac {1}{2}}} \left ( {{\sl Y}_{{\frac {1}{2}\sqrt {{{\it a1}}^{2}-2\,{\it a1}-4\,{\it a2}+1}}}\left (\sqrt {{\it b2}}x\right )}{\it \_C2}+{{\sl J}_{{\frac {1}{2}\sqrt {{{\it a1}}^{2}-2\,{\it a1}-4\,{\it a2}+1}}}\left (\sqrt {{\it b2}}x\right )}{\it \_C1} \right ) \right \} \] Mathematica raw input

DSolve[(a2 + b2*x^2)*y[x] + a1*x*y'[x] + x^2*y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> x^(1/2 - a1/2)*(BesselJ[Sqrt[1 - 2*a1 + a1^2 - 4*a2]/2, Sqrt[b2]*x]*C[
1] + BesselY[Sqrt[1 - 2*a1 + a1^2 - 4*a2]/2, Sqrt[b2]*x]*C[2])}}

Maple raw input

dsolve(x^2*diff(diff(y(x),x),x)+a1*x*diff(y(x),x)+(b2*x^2+a2)*y(x) = 0, y(x),'implicit')

Maple raw output

y(x) = x^(-1/2*a1+1/2)*(BesselY(1/2*(a1^2-2*a1-4*a2+1)^(1/2),b2^(1/2)*x)*_C2+Bes
selJ(1/2*(a1^2-2*a1-4*a2+1)^(1/2),b2^(1/2)*x)*_C1)