Internal problem ID [8628]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1048.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-4 y^{\prime } x +\left (3 x^{2}+2 n -1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.105 (sec). Leaf size: 43
dsolve(diff(diff(y(x),x),x)-4*x*diff(y(x),x)+(3*x^2+2*n-1)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{\frac {x^{2}}{2}} \KummerM \left (-\frac {n}{2}+\frac {1}{2}, \frac {3}{2}, x^{2}\right ) x +c_{2} {\mathrm e}^{\frac {x^{2}}{2}} \KummerU \left (-\frac {n}{2}+\frac {1}{2}, \frac {3}{2}, x^{2}\right ) x \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 37
DSolve[(-1 + 2*n + 3*x^2)*y[x] - 4*x*y'[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{\frac {x^2}{2}} \left (c_1 \text {HermiteH}(n,x)+c_2 \, _1F_1\left (-\frac {n}{2};\frac {1}{2};x^2\right )\right ) \\ \end{align*}