Internal problem ID [8857]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1277.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {4 y^{\prime \prime } x^{2}+4 x y^{\prime }-\left (a \,x^{2}+1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.011 (sec). Leaf size: 29
dsolve(4*x^2*diff(diff(y(x),x),x)+4*x*diff(y(x),x)-(a*x^2+1)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \sinh \left (\frac {x \sqrt {a}}{2}\right )}{\sqrt {x}}+\frac {c_{2} \cosh \left (\frac {x \sqrt {a}}{2}\right )}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.018 (sec). Leaf size: 49
DSolve[(-1 - a*x^2)*y[x] + 4*x*y'[x] + 4*x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{-\frac {\sqrt {a} x}{2}} \left (c_2 e^{\sqrt {a} x}+\sqrt {a} c_1\right )}{\sqrt {a} \sqrt {x}} \\ \end{align*}