Internal problem ID [8617]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1037.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+a y^{\prime }-\left (b^{2} x^{2}+c \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.022 (sec). Leaf size: 73
dsolve(diff(diff(y(x),x),x)+a*diff(y(x),x)-(b^2*x^2+c)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} x \KummerM \left (\frac {a^{2}+12 b +4 c}{16 b}, \frac {3}{2}, b \,x^{2}\right ) {\mathrm e}^{-\frac {x \left (b x +a \right )}{2}}+c_{2} x \KummerU \left (\frac {a^{2}+12 b +4 c}{16 b}, \frac {3}{2}, b \,x^{2}\right ) {\mathrm e}^{-\frac {x \left (b x +a \right )}{2}} \]
✓ Solution by Mathematica
Time used: 0.014 (sec). Leaf size: 74
DSolve[(-c - b^2*x^2)*y[x] + a*y'[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-\frac {1}{2} x (a+b x)} \left (c_1 \text {HermiteH}\left (-\frac {a^2+4 (b+c)}{8 b},\sqrt {b} x\right )+c_2 \, _1F_1\left (\frac {a^2+4 (b+c)}{16 b};\frac {1}{2};b x^2\right )\right ) \\ \end{align*}