Internal problem ID [10924]
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: 90.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x y^{\prime \prime }+\left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime }+\left (d -1\right ) \left (a \,x^{2}+b x +c \right ) y=0} \]
✓ Solution by Maple
Time used: 0.046 (sec). Leaf size: 48
dsolve(x*diff(y(x),x$2)+(a*x^3+b*x^2+c*x+d)*diff(y(x),x)+(d-1)*(a*x^2+b*x+c)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} x^{-d +1}+c_{2} x^{-d +1} \left (\int x^{d -2} {\mathrm e}^{-\frac {\left (a \,x^{2}+\frac {3}{2} b x +3 c \right ) x}{3}}d x \right ) \]
✓ Solution by Mathematica
Time used: 1.839 (sec). Leaf size: 57
DSolve[x*y''[x]+(a*x^3+b*x^2+c*x+d)*y'[x]+(d-1)*(a*x^2+b*x+c)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to x^{1-d} \left (c_2 \int _1^x\exp \left (-\frac {1}{6} K[1] (6 c+K[1] (3 b+2 a K[1]))\right ) K[1]^{d-2}dK[1]+c_1\right ) \]