Internal problem ID [8634]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1054.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (c x +d \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.007 (sec). Leaf size: 105
dsolve(diff(diff(y(x),x),x)+(a*x+b)*diff(y(x),x)+(c*x+d)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{-\frac {c x}{a}} \KummerM \left (\frac {d \,a^{2}-a b c +c^{2}}{2 a^{3}}, \frac {1}{2}, -\frac {\left (a^{2} x +a b -2 c \right )^{2}}{2 a^{3}}\right )+c_{2} {\mathrm e}^{-\frac {c x}{a}} \KummerU \left (\frac {d \,a^{2}-a b c +c^{2}}{2 a^{3}}, \frac {1}{2}, -\frac {\left (a^{2} x +a b -2 c \right )^{2}}{2 a^{3}}\right ) \]
✓ Solution by Mathematica
Time used: 0.023 (sec). Leaf size: 130
DSolve[(d + c*x)*y[x] + (b + a*x)*y'[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{\frac {c x}{a}-\frac {a x^2}{2}-b x} \left (c_1 \text {HermiteH}\left (\frac {-a^3+a^2 d-a b c+c^2}{a^3},\frac {a (a x+b)-2 c}{\sqrt {2} a^{3/2}}\right )+c_2 \, _1F_1\left (\frac {a^3-d a^2+b c a-c^2}{2 a^3};\frac {1}{2};\frac {(a (b+a x)-2 c)^2}{2 a^3}\right )\right ) \\ \end{align*}