61.28.9 problem 69

Internal problem ID [12569]
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
Problem number : 69
Date solved : Tuesday, January 28, 2025 at 03:21:51 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} x y^{\prime \prime }+a x y^{\prime }+a y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 29

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

\[ y = \operatorname {Ei}_{1}\left (-a x \right ) c_{1} a x \,{\mathrm e}^{-a x}+c_{1} +c_{2} x \,{\mathrm e}^{-a x} \]

✓ Solution by Mathematica

Time used: 0.275 (sec). Leaf size: 37

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

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