Internal problem ID [10105]
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: 24.
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 )-a y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 66
dsolve(diff(y(x),x$2)+(a*x+b)*diff(y(x),x)-a*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \left (x a +b \right )+c_{2} \left (\pi \left (x a +b \right ) {\mathrm e}^{\frac {b^{2}}{2 a}} \erf \left (\frac {\sqrt {2}\, \left (x a +b \right )}{2 \sqrt {a}}\right )+\sqrt {2}\, \sqrt {\pi }\, \sqrt {a}\, {\mathrm e}^{-\frac {x \left (x a +2 b \right )}{2}}\right ) \]
✓ Solution by Mathematica
Time used: 0.274 (sec). Leaf size: 82
DSolve[y''[x]+(a*x+b)*y'[x]-a*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {(a x+b) \left (-\frac {\sqrt {\frac {\pi }{2}} c_2 \text {Erf}\left (\frac {a x+b}{\sqrt {2} \sqrt {a}}\right )}{a^{3/2}}-\frac {c_2 e^{-\frac {(a x+b)^2}{2 a}}}{a (a x+b)}+c_1\right )}{b} \\ \end{align*}