Internal problem ID [10915]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-3 Equation of form
\((a x + b)y''+f(x)y'+g(x)y=0\)
Problem number: 81.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x y^{\prime \prime }+\left (a \,x^{2}+b x +2\right ) y^{\prime }+b y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 72
dsolve(x*diff(y(x),x$2)+(a*x^2+b*x+2)*diff(y(x),x)+b*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {c_{1} \left (a x +b \right )}{x}+\frac {c_{2} \left (\pi \left (a x +b \right ) {\mathrm e}^{\frac {b^{2}}{2 a}} \operatorname {erf}\left (\frac {\sqrt {2}\, \left (a x +b \right )}{2 \sqrt {a}}\right )+{\mathrm e}^{-\frac {x \left (a x +2 b \right )}{2}} \sqrt {2}\, \sqrt {\pi }\, \sqrt {a}\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.535 (sec). Leaf size: 85
DSolve[x*y''[x]+(a*x^2+b*x+2)*y'[x]+b*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {(a x+b) \left (-\frac {\sqrt {\frac {\pi }{2}} c_2 \text {erf}\left (\frac {a x+b}{\sqrt {2} \sqrt {a}}\right )}{a^{3/2}}-\frac {c_2 e^{-\frac {(a x+b)^2}{2 a}}}{a (a x+b)}+c_1\right )}{b x} \]