61.24.6 problem 6

Internal problem ID [12419]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.3. Abel Equations of the Second Kind. subsection 1.3.3-2.
Problem number : 6
Date solved : Tuesday, January 28, 2025 at 07:58:26 PM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 27

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

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

✓ Solution by Mathematica

Time used: 0.128 (sec). Leaf size: 30

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

\[ \text {Solve}\left [\operatorname {ExpIntegralEi}\left (x+\frac {y(x)}{a}\right )+c_1=\frac {e^{\frac {y(x)}{a}+x}}{x},y(x)\right ] \]