Internal problem ID [8750]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1170.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+2 x y^{\prime }+\left (a \,x^{2}+b \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.01 (sec). Leaf size: 45
dsolve(x^2*diff(diff(y(x),x),x)+2*x*diff(y(x),x)+(a*x^2+b)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \BesselJ \left (\frac {\sqrt {1-4 b}}{2}, x \sqrt {a}\right )}{\sqrt {x}}+\frac {c_{2} \BesselY \left (\frac {\sqrt {1-4 b}}{2}, x \sqrt {a}\right )}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.01 (sec). Leaf size: 58
DSolve[(b + a*x^2)*y[x] + 2*x*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 j_{\frac {1}{2} \left (\sqrt {1-4 b}-1\right )}\left (\sqrt {a} x\right )+c_2 y_{\frac {1}{2} \left (\sqrt {1-4 b}-1\right )}\left (\sqrt {a} x\right ) \\ \end{align*}