61.27.15 problem 25

Internal problem ID [12525]
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 : 25
Date solved : Tuesday, January 28, 2025 at 03:19:23 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 36

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

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

✓ Solution by Mathematica

Time used: 60.037 (sec). Leaf size: 49

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

\[ y(x)\to e^{-\frac {1}{2} x (a x+2 b)} \left (\int _1^xe^{\frac {1}{2} a K[1]^2+b K[1]} c_1dK[1]+c_2\right ) \]