Internal problem ID [10850]
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: 26.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c \left (a x +b -c \right ) y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 31
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)
\[ y \left (x \right ) = {\mathrm e}^{-c x} \left (c_{1} +\operatorname {erf}\left (\frac {\sqrt {2}\, \left (a x +b -2 c \right )}{2 \sqrt {a}}\right ) c_{2} \right ) \]
✓ Solution by Mathematica
Time used: 0.091 (sec). Leaf size: 70
DSolve[y''[x]+(a*x+b)*y'[x]+c*(a*x+b-c)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-\frac {1}{2} x (a x+2 b-2 c)} \left (c_1 \operatorname {HermiteH}\left (-1,\frac {b-2 c+a x}{\sqrt {2} \sqrt {a}}\right )+c_2 e^{\frac {(a x+b-2 c)^2}{2 a}}\right ) \]