Internal problem ID [8669]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1089.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {a y^{\prime \prime }-\left (a b +c +x \right ) y^{\prime }+\left (b \left (x +c \right )+d \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.008 (sec). Leaf size: 61
dsolve(a*diff(diff(y(x),x),x)-(a*b+c+x)*diff(y(x),x)+(b*(x+c)+d)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{b x} \KummerM \left (-\frac {d}{2}, \frac {1}{2}, \frac {\left (a b -c -x \right )^{2}}{2 a}\right )+c_{2} {\mathrm e}^{b x} \KummerU \left (-\frac {d}{2}, \frac {1}{2}, \frac {\left (a b -c -x \right )^{2}}{2 a}\right ) \]
✓ Solution by Mathematica
Time used: 0.022 (sec). Leaf size: 63
DSolve[(d + b*(c + x))*y[x] - (a*b + c + x)*y'[x] + a*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{b x} \left (c_1 H_d\left (\frac {-a b+c+x}{\sqrt {2} \sqrt {a}}\right )+c_2 \, _1F_1\left (-\frac {d}{2};\frac {1}{2};\frac {(-a b+c+x)^2}{2 a}\right )\right ) \\ \end{align*}