Internal problem ID [8593]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1013.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-\left (a^{2} x^{2}+a \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 29
dsolve(diff(diff(y(x),x),x)-(a^2*x^2+a)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{\frac {a \,x^{2}}{2}}+c_{2} {\mathrm e}^{\frac {a \,x^{2}}{2}} \erf \left (x \sqrt {a}\right ) \]
✓ Solution by Mathematica
Time used: 0.008 (sec). Leaf size: 36
DSolve[(-a - a^2*x^2)*y[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 D_{-1}\left (\sqrt {2} \sqrt {a} x\right )+c_2 e^{\frac {a x^2}{2}} \\ \end{align*}