61.22.1 problem 1

Internal problem ID [12326]
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 1.3.1-2. Solvable equations and their solutions
Problem number : 1
Date solved : Tuesday, January 28, 2025 at 02:46:46 AM
CAS classification : [_quadrature]

\begin{align*} y^{\prime } y-y&=A \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 30

dsolve(y(x)*diff(y(x),x)-y(x)=A,y(x), singsol=all)
 

\[ y = -A \left (\operatorname {LambertW}\left (-\frac {{\mathrm e}^{\frac {-A -c_{1} -x}{A}}}{A}\right )+1\right ) \]

✓ Solution by Mathematica

Time used: 0.126 (sec). Leaf size: 35

DSolve[y[x]*D[y[x],x]-y[x]==A,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {K[1]}{A+K[1]}dK[1]\&\right ][x+c_1] \\ y(x)\to -A \\ \end{align*}