61.24.7 problem 7

Internal problem ID [12420]
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 : 7
Date solved : Tuesday, January 28, 2025 at 07:58:27 PM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

✓ Solution by Maple

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

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

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

✓ Solution by Mathematica

Time used: 0.181 (sec). Leaf size: 45

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

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