Internal problem ID [10651]
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`]]
\[ \boxed {y y^{\prime }-y=A x +B} \]
✓ Solution by Maple
Time used: 4.266 (sec). Leaf size: 68
dsolve(y(x)*diff(y(x),x)-y(x)=A*x+B,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {\left (x A +B \right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-A +\textit {\_Z} +{\mathrm e}^{\operatorname {RootOf}\left (\left (x A +B \right )^{2} \left (-2 \,{\mathrm e}^{\textit {\_Z}} \cosh \left (\left (\textit {\_Z} +2 \ln \left (x A +B \right )+2 c_{1} \right ) \sqrt {4 A +1}\right )+4 A -2 \,{\mathrm e}^{\textit {\_Z}}+1\right )\right )}\right )}{A} \]
✓ Solution by Mathematica
Time used: 0.184 (sec). Leaf size: 88
DSolve[y[x]*y'[x]-y[x]==A*x+B,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\frac {\frac {2 \arctan \left (\frac {\frac {2 A y(x)}{A x+B}-1}{\sqrt {-4 A-1}}\right )}{\sqrt {-4 A-1}}+\log \left (-\frac {A y(x)^2}{(A x+B)^2}+\frac {y(x)}{A x+B}+1\right )}{2 A}=\frac {\log (A x+B)}{A}+c_1,y(x)\right ] \]