Internal problem ID [8790]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1210.
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}-2 x \left (x^{2}-a \right ) y^{\prime }+\left (2 n \,x^{2}+\left (\left (-1\right )^{n}-1\right ) a \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.406 (sec). Leaf size: 93
dsolve(x^2*diff(diff(y(x),x),x)-2*x*(x^2-a)*diff(y(x),x)+(2*n*x^2+((-1)^n-1)*a)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{\frac {x^{2}}{2}} x^{-\frac {1}{2}-a} \WhittakerM \left (\frac {a}{2}+\frac {n}{2}+\frac {1}{4}, \frac {\sqrt {1-4 a \left (-1\right )^{n}+4 a^{2}}}{4}, x^{2}\right )+c_{2} {\mathrm e}^{\frac {x^{2}}{2}} x^{-\frac {1}{2}-a} \WhittakerW \left (\frac {a}{2}+\frac {n}{2}+\frac {1}{4}, \frac {\sqrt {1-4 a \left (-1\right )^{n}+4 a^{2}}}{4}, x^{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.131 (sec). Leaf size: 231
DSolve[((-1 + (-1)^n)*a + 2*n*x^2)*y[x] - 2*x*(-a + x^2)*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to i^{-a} (-1)^{\frac {1}{4} \left (1-\sqrt {4 a^2-4 a (-1)^n+1}\right )} x^{\frac {1}{2} \left (-\sqrt {4 a^2-4 a (-1)^n+1}-2 a+1\right )} \left (c_1 \, _1F_1\left (\frac {1}{4} \left (-2 a-2 n-\sqrt {4 a^2-4 (-1)^n a+1}+1\right );1-\frac {1}{2} \sqrt {4 a^2-4 (-1)^n a+1};x^2\right )+c_2 i^{\sqrt {4 a^2-4 a (-1)^n+1}} x^{\sqrt {4 a^2-4 a (-1)^n+1}} \, _1F_1\left (\frac {1}{4} \left (-2 a-2 n+\sqrt {4 a^2-4 (-1)^n a+1}+1\right );\frac {1}{2} \left (\sqrt {4 a^2-4 (-1)^n a+1}+2\right );x^2\right )\right ) \\ \end{align*}