Internal problem ID [10106]
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: 25.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+a y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 52
dsolve(diff(y(x),x$2)+(a*x+b)*diff(y(x),x)+a*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \erf \left (-\frac {\sqrt {-2 a}\, x}{2}+\frac {b}{\sqrt {-2 a}}\right ) {\mathrm e}^{-\frac {1}{2} a \,x^{2}-b x} c_{1}+c_{2} {\mathrm e}^{-\frac {1}{2} a \,x^{2}-b x} \]
✓ Solution by Mathematica
Time used: 0.036 (sec). Leaf size: 53
DSolve[y''[x]+(a*x+b)*y'[x]+a*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {2} c_1 F\left (\frac {b+a x}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {a}}+c_2 e^{-\frac {1}{2} x (a x+2 b)} \\ \end{align*}