Internal problem ID [8794]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1214.
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}+\left (\left (-1\right )^{n} a -x^{4}+\left (2 a +2 n +1\right ) x^{2}-a^{2}\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 73
dsolve(x^2*diff(diff(y(x),x),x)+(-x^4+(2*n+2*a+1)*x^2+(-1)^n*a-a^2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \WhittakerM \left (\frac {a}{2}+\frac {n}{2}+\frac {1}{4}, \frac {\sqrt {1-4 a \left (-1\right )^{n}+4 a^{2}}}{4}, x^{2}\right )}{\sqrt {x}}+\frac {c_{2} \WhittakerW \left (\frac {a}{2}+\frac {n}{2}+\frac {1}{4}, \frac {\sqrt {1-4 a \left (-1\right )^{n}+4 a^{2}}}{4}, x^{2}\right )}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.146 (sec). Leaf size: 191
DSolve[((-1)^n*a - a^2 + (1 + 2*a + 2*n)*x^2 - x^4)*y[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{-\frac {x^2}{2}} 2^{\frac {1}{4} \left (\sqrt {4 a^2-4 a (-1)^n+1}+2\right )} \left (x^2\right )^{\frac {1}{4} \left (\sqrt {4 a^2-4 a (-1)^n+1}+2\right )} \left (c_1 \text {HypergeometricU}\left (\frac {1}{4} \left (\sqrt {4 a^2-4 a (-1)^n+1}-2 a-2 n+1\right ),\frac {1}{2} \left (\sqrt {4 a^2-4 a (-1)^n+1}+2\right ),x^2\right )+c_2 L_{\frac {1}{4} \left (2 a+2 n-\sqrt {4 a^2-4 (-1)^n a+1}-1\right )}^{\frac {1}{2} \sqrt {4 a^2-4 (-1)^n a+1}}\left (x^2\right )\right )}{\sqrt {x}} \\ \end{align*}