Internal problem ID [10107]
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: 26.
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 } \left (x a +b \right )+c \left (x a +b -c \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 36
dsolve(diff(y(x),x$2)+(a*x+b)*diff(y(x),x)+c*(a*x+b-c)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{-c x}+c_{2} {\mathrm e}^{-c x} \erf \left (\frac {\sqrt {2}\, \left (x a +b -2 c \right )}{2 \sqrt {a}}\right ) \]
✓ Solution by Mathematica
Time used: 0.015 (sec). Leaf size: 63
DSolve[y''[x]+(a*x+b)*y'[x]+c*(a*x+b-c)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} e^{\frac {(b-2 c)^2}{2 a}-c x} \left (\sqrt {\pi } c_1 \text {Erfc}\left (\frac {a x+b-2 c}{\sqrt {2} \sqrt {a}}\right )+2 c_2\right ) \\ \end{align*}