Internal problem ID [8749]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1169.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime } x^{2}+2 x y^{\prime }+\left (a x -b^{2}\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 51
dsolve(x^2*diff(diff(y(x),x),x)+2*x*diff(y(x),x)+(a*x-b^2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \BesselJ \left (\sqrt {4 b^{2}+1}, 2 \sqrt {a}\, \sqrt {x}\right )}{\sqrt {x}}+\frac {c_{2} \BesselY \left (\sqrt {4 b^{2}+1}, 2 \sqrt {a}\, \sqrt {x}\right )}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.036 (sec). Leaf size: 103
DSolve[(-b^2 + a*x)*y[x] + 2*x*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_1 \text {Gamma}\left (1-\sqrt {4 b^2+1}\right ) J_{-\sqrt {4 b^2+1}}\left (2 \sqrt {a} \sqrt {x}\right )+c_2 \text {Gamma}\left (\sqrt {4 b^2+1}+1\right ) J_{\sqrt {4 b^2+1}}\left (2 \sqrt {a} \sqrt {x}\right )}{\sqrt {a} \sqrt {x}} \\ \end{align*}