Internal problem ID [8626]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1046.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-2 y^{\prime } x +a y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.022 (sec). Leaf size: 31
dsolve(diff(diff(y(x),x),x)-2*x*diff(y(x),x)+a*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \KummerM \left (\frac {1}{2}-\frac {a}{4}, \frac {3}{2}, x^{2}\right ) x +c_{2} \KummerU \left (\frac {1}{2}-\frac {a}{4}, \frac {3}{2}, x^{2}\right ) x \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 31
DSolve[a*y[x] - 2*x*y'[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \text {HermiteH}\left (\frac {a}{2},x\right )+c_2 \, _1F_1\left (-\frac {a}{4};\frac {1}{2};x^2\right ) \\ \end{align*}