Internal problem ID [10103]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-1 Equation of form
\(y''+f(x)y'+g(x)y=0\)
Problem number: 22.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+y^{\prime } a x +\left (b x +c \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.049 (sec). Leaf size: 89
dsolve(diff(y(x),x$2)+a*x*diff(y(x),x)+(b*x+c)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{-\frac {b x}{a}} \KummerM \left (\frac {c \,a^{2}+b^{2}}{2 a^{3}}, \frac {1}{2}, -\frac {\left (a^{2} x -2 b \right )^{2}}{2 a^{3}}\right )+c_{2} {\mathrm e}^{-\frac {b x}{a}} \KummerU \left (\frac {c \,a^{2}+b^{2}}{2 a^{3}}, \frac {1}{2}, -\frac {\left (a^{2} x -2 b \right )^{2}}{2 a^{3}}\right ) \]
✓ Solution by Mathematica
Time used: 0.012 (sec). Leaf size: 107
DSolve[y''[x]+a*x*y'[x]+(b*x+c)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{\frac {b x}{a}-\frac {a x^2}{2}} \left (c_1 \text {HermiteH}\left (\frac {b^2}{a^3}+\frac {c}{a}-1,\frac {a^2 x-2 b}{\sqrt {2} a^{3/2}}\right )+c_2 \, _1F_1\left (-\frac {(c-a) a^2+b^2}{2 a^3};\frac {1}{2};\frac {\left (a^2 x-2 b\right )^2}{2 a^3}\right )\right ) \\ \end{align*}