Internal problem ID [10859]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-2 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]]
\[ \boxed {y^{\prime \prime }+y^{\prime } \left (a x +b \right )+a y=0} \]
✓ Solution by Maple
Time used: 0.0 (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)
\[ y \left (x \right ) = \operatorname {erf}\left (-\frac {\sqrt {-2 a}\, x}{2}+\frac {b}{\sqrt {-2 a}}\right ) {\mathrm e}^{-\frac {1}{2} a \,x^{2}-x b} c_{1} +c_{2} {\mathrm e}^{-\frac {1}{2} a \,x^{2}-x b} \]
✓ Solution by Mathematica
Time used: 0.067 (sec). Leaf size: 79
DSolve[y''[x]+(a*x+b)*y'[x]+a*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {e^{-\frac {(a x+b)^2}{2 a}} \left (2 \sqrt {a} c_2 e^{\frac {b^2}{2 a}}+\sqrt {2 \pi } c_1 \text {erfi}\left (\frac {a x+b}{\sqrt {2} \sqrt {a}}\right )\right )}{2 \sqrt {a}} \]