Internal problem ID [9908]
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. Form \(y y'-y=f(x)\). subsection 1.3.1-2. Solvable
equations and their solutions
Problem number: 2.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class C], _rational, [_Abel, 2nd type, class A]]
Solve \begin {gather*} \boxed {y^{\prime } y-y-x A -B=0} \end {gather*}
✓ Solution by Maple
Time used: 0.593 (sec). Leaf size: 119
dsolve(y(x)*diff(y(x),x)-y(x)=A*x+B,y(x), singsol=all)
\[ y \relax (x ) = -\frac {\RootOf \left (\textit {\_Z}^{2}-A +{\mathrm e}^{\RootOf \left (\left (x A +B \right )^{2} \left (4 \left (\tanh ^{2}\left (\ln \left (x A +B \right ) \sqrt {4 A +1}+\frac {\textit {\_Z} \sqrt {4 A +1}}{2}+c_{1} \sqrt {4 A +1}\right )\right ) A +\tanh ^{2}\left (\ln \left (x A +B \right ) \sqrt {4 A +1}+\frac {\textit {\_Z} \sqrt {4 A +1}}{2}+c_{1} \sqrt {4 A +1}\right )-4 A +4 \,{\mathrm e}^{\textit {\_Z}}-1\right )\right )}+\textit {\_Z} \right ) \left (x A +B \right )}{A} \]
✓ Solution by Mathematica
Time used: 0.114 (sec). Leaf size: 88
DSolve[y[x]*y'[x]-y[x]==A*x+B,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\frac {\frac {2 \text {ArcTan}\left (\frac {\frac {2 A y(x)}{A x+B}-1}{\sqrt {-4 A-1}}\right )}{\sqrt {-4 A-1}}+\log \left (-\frac {A y(x)^2}{(A x+B)^2}+\frac {y(x)}{A x+B}+1\right )}{2 A}=\frac {\log (A x+B)}{A}+c_1,y(x)\right ] \]