Internal problem ID [8667]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1087.
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 }-\left (x^{2}+a \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.027 (sec). Leaf size: 35
dsolve(4*diff(diff(y(x),x),x)-(x^2+a)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \WhittakerM \left (-\frac {a}{8}, \frac {1}{4}, \frac {x^{2}}{2}\right )}{\sqrt {x}}+\frac {c_{2} \WhittakerW \left (-\frac {a}{8}, \frac {1}{4}, \frac {x^{2}}{2}\right )}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 36
DSolve[(-a - x^2)*y[x] + 4*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 D_{\frac {1}{4} (-a-2)}(x)+c_2 D_{\frac {a-2}{4}}(i x) \\ \end{align*}