61.22.2 problem 2

Internal problem ID [12327]
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 : 2
Date solved : Tuesday, January 28, 2025 at 02:46:47 AM
CAS classification : [[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

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

✓ Solution by Maple

Time used: 1.990 (sec). Leaf size: 68

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

\[ y = -\frac {\left (x A +B \right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-A +\textit {\_Z} +{\mathrm e}^{\operatorname {RootOf}\left (\left (x A +B \right )^{2} \left (-2 \,{\mathrm e}^{\textit {\_Z}} \cosh \left (\left (\textit {\_Z} +2 \ln \left (x A +B \right )+2 c_{1} \right ) \sqrt {4 A +1}\right )+4 A -2 \,{\mathrm e}^{\textit {\_Z}}+1\right )\right )}\right )}{A} \]

✓ Solution by Mathematica

Time used: 0.038 (sec). Leaf size: 46

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

\[ \text {Solve}\left [\int _1^{\frac {y(x)}{B+A x}}\frac {1}{-A K[1]+1+\frac {1}{K[1]}}dK[1]=\frac {\log (A x+B)}{A}+c_1,y(x)\right ] \]