Internal problem ID [8592]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1012.
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 (x^{2}+a \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 31
dsolve(diff(diff(y(x),x),x)-(x^2+a)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \WhittakerM \left (-\frac {a}{4}, \frac {1}{4}, x^{2}\right )}{\sqrt {x}}+\frac {c_{2} \WhittakerW \left (-\frac {a}{4}, \frac {1}{4}, x^{2}\right )}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 47
DSolve[(-a - x^2)*y[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 D_{\frac {1}{2} (-a-1)}\left (\sqrt {2} x\right )+c_2 D_{\frac {a-1}{2}}\left (i \sqrt {2} x\right ) \\ \end{align*}