27.4 problem 14

Internal problem ID [10095]

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: 14.
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 \left (-b \,x^{2}+x a +1\right ) y=0} \end {gather*}

Solution by Maple

Time used: 0.0 (sec). Leaf size: 40

dsolve(diff(y(x),x$2)+a*diff(y(x),x)+b*(-b*x^2+a*x+1)*y(x)=0,y(x), singsol=all)
 

\[ y \relax (x ) = \erf \left (\sqrt {-b}\, x +\frac {a}{2 \sqrt {-b}}\right ) {\mathrm e}^{-\frac {b \,x^{2}}{2}} c_{1}+c_{2} {\mathrm e}^{-\frac {b \,x^{2}}{2}} \]

Solution by Mathematica

Time used: 0.086 (sec). Leaf size: 56

DSolve[y''[x]+a*y'[x]+b*(-b*x^2+a*x+1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to e^{-\frac {b x^2}{2}} \left (\frac {c_2 e^{x (b x-a)} F\left (\frac {2 b x-a}{2 \sqrt {b}}\right )}{\sqrt {b}}+c_1\right ) \\ \end{align*}