Internal problem ID [10906]
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: 72.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x y^{\prime \prime }+\left (2 a x +b \right ) y^{\prime }+a \left (a x +b \right ) y=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 26
dsolve(x*diff(y(x),x$2)+(2*a*x+b)*diff(y(x),x)+a*(a*x+b)*y(x)=0,y(x), singsol=all)
\[ y = c_{1} {\mathrm e}^{-a x}+x^{-b +1} c_{2} {\mathrm e}^{-a x} \]
✓ Solution by Mathematica
Time used: 0.233 (sec). Leaf size: 70
DSolve[x*y''[x]+(2*a*x+b)*y'[x]+a*(a*x+b)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {e^{-a x} x^{\frac {1}{2} \left (-b-\sqrt {(b-1)^2}+1\right )} \left (c_2 x^{\sqrt {(b-1)^2}}+\sqrt {(b-1)^2} c_1\right )}{\sqrt {(b-1)^2}} \]