Internal problem ID [10923]
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: 89.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x y^{\prime \prime }+\left (b \,x^{3} a +b \,x^{2}+a x -1\right ) y^{\prime }+a^{2} b \,x^{3} y=0} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 58
dsolve(x*diff(y(x),x$2)+(a*b*x^3+b*x^2+a*x-1)*diff(y(x),x)+a^2*b*x^3*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{-a x} \left (a x +1\right )+c_{2} {\mathrm e}^{-a x} \left (a x +1\right ) \left (\int \frac {x \,{\mathrm e}^{-\frac {x \left (a b \,x^{2}+\frac {3}{2} b x -3 a \right )}{3}}}{\left (a x +1\right )^{2}}d x \right ) \]
✓ Solution by Mathematica
Time used: 4.606 (sec). Leaf size: 72
DSolve[x*y''[x]+(a*b*x^3+b*x^2+a*x-1)*y'[x]+a^2*b*x^3*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {e^{-a x} (a x+1) \left (c_2 \int _1^x\frac {a^2 \exp \left (-\frac {1}{6} K[1] \left (3 b K[1]+2 a \left (b K[1]^2-3\right )\right )\right ) K[1]}{(a K[1]+1)^2}dK[1]+c_1\right )}{a} \]