Internal problem ID [10096]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-1 Equation of form
\(y''+f(x)y'+g(x)y=0\)
Problem number: 15.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, _with_symmetry_[0,F(x)]]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+a y^{\prime }+b x \left (-b \,x^{3}+x a +2\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 33
dsolve(diff(y(x),x$2)+a*diff(y(x),x)+b*x*(-b*x^3+a*x+2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \left (\left (\int {\mathrm e}^{-x a} {\mathrm e}^{\frac {2 x^{3} b}{3}}d x \right ) c_{1}+c_{2}\right ) {\mathrm e}^{-\frac {x^{3} b}{3}} \]
✓ Solution by Mathematica
Time used: 0.132 (sec). Leaf size: 46
DSolve[y''[x]+a*y'[x]+b*x*(-b*x^3+a*x+2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-\frac {b x^3}{3}} \left (c_2 \int _1^xe^{\frac {2}{3} b K[1]^3-a K[1]}dK[1]+c_1\right ) \\ \end{align*}