61.27.4 problem 14

Internal problem ID [12514]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-2
Problem number : 14
Date solved : Tuesday, January 28, 2025 at 03:19:11 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} y^{\prime \prime }+a y^{\prime }+b \left (-b \,x^{2}+a x +1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 30

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)
 

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

✓ Solution by Mathematica

Time used: 0.477 (sec). Leaf size: 42

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

\[ y(x)\to e^{-\frac {b x^2}{2}} \left (c_2 \int _1^xe^{K[1] (b K[1]-a)}dK[1]+c_1\right ) \]