61.23.3 problem 3

Internal problem ID [12404]
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 1.3.2.
Problem number : 3
Date solved : Tuesday, January 28, 2025 at 07:58:09 PM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 39

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

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

✓ Solution by Mathematica

Time used: 0.166 (sec). Leaf size: 37

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

\[ \text {Solve}\left [\operatorname {ExpIntegralEi}(a (a x-y(x)))+c_1=\frac {e^{a (a x-y(x))}}{a^2 x},y(x)\right ] \]