Internal problem ID [10921]
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: 87.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
\[ \boxed {x y^{\prime \prime }+x \left (a \,x^{2}+b \right ) y^{\prime }+\left (3 a \,x^{2}+b \right ) y=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 42
dsolve(x*diff(y(x),x$2)+x*(a*x^2+b)*diff(y(x),x)+(3*a*x^2+b)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (c_{1} \left (\int \frac {{\mathrm e}^{\frac {a \,x^{3}}{3}} {\mathrm e}^{b x}}{x^{2}}d x \right )+c_{2} \right ) {\mathrm e}^{-\frac {a \,x^{3}}{3}} {\mathrm e}^{-b x} x \]
✓ Solution by Mathematica
Time used: 2.43 (sec). Leaf size: 56
DSolve[x*y''[x]+x*(a*x^2+b)*y'[x]+(3*a*x^2+b)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to x e^{-\frac {a x^3}{3}-b x} \left (c_2 \int _1^x\frac {e^{\frac {1}{3} a K[1]^3+b K[1]}}{K[1]^2}dK[1]+c_1\right ) \]