Internal problem ID [10914]
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: 80.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }-\left (a c \,x^{2}+\left (b c +c^{2}+a \right ) x +b +2 c \right ) y=0} \]
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 39
dsolve(x*diff(y(x),x$2)+(a*x^2+b*x)*diff(y(x),x)-(a*c*x^2+(a+b*c+c^2)*x+b+2*c)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{c x} x +c_{2} {\mathrm e}^{c x} x \left (\int \frac {{\mathrm e}^{-\frac {x \left (a x +2 b +4 c \right )}{2}}}{x^{2}}d x \right ) \]
✓ Solution by Mathematica
Time used: 3.129 (sec). Leaf size: 49
DSolve[x*y''[x]+(a*x^2+b*x)*y'[x]-(a*c*x^2+(a+b*c+c^2)*x+b+2*c)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to x e^{c x} \left (c_2 \int _1^x\frac {e^{-\frac {1}{2} K[1] (2 b+4 c+a K[1])}}{K[1]^2}dK[1]+c_1\right ) \]