Internal problem ID [9923]
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: 17.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_Abel, 2nd type, class B]]
Solve \begin {gather*} \boxed {y^{\prime } y-y+\frac {x}{4}-\frac {A \left (\sqrt {x}+5 A +\frac {3 A^{2}}{\sqrt {x}}\right )}{4}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 254
dsolve(y(x)*diff(y(x),x)-y(x)=-1/4*x+1/4*A*(x^(1/2)+5*A+3*A^2*x^(-1/2)),y(x), singsol=all)
\[ c_{1}+\frac {2 A \left (\int _{}^{\frac {6 A \sqrt {x}-2 x +3 y \relax (x )}{12 A^{2}-4 A \sqrt {x}+2 y \relax (x )}}\frac {{\mathrm e}^{-\frac {2}{2 \textit {\_a} +1}} \sqrt {2 \textit {\_a} +1}}{\sqrt {2 \textit {\_a} -3}}d \textit {\_a} \right ) \left (6 A^{2}-2 A \sqrt {x}+y \relax (x )\right ) \sqrt {-\frac {\left (3 A -\sqrt {x}\right )^{2}}{6 A^{2}-2 A \sqrt {x}+y \relax (x )}}-y \relax (x ) {\mathrm e}^{\frac {-6 A^{2}+2 A \sqrt {x}-y \relax (x )}{3 A^{2}+2 A \sqrt {x}-x +2 y \relax (x )}} \sqrt {\frac {3 A^{2}+2 A \sqrt {x}-x +2 y \relax (x )}{6 A^{2}-2 A \sqrt {x}+y \relax (x )}}\, \left (3 A -\sqrt {x}\right )}{\sqrt {-\frac {\left (3 A -\sqrt {x}\right )^{2}}{6 A^{2}-2 A \sqrt {x}+y \relax (x )}}\, \left (6 A^{2}-2 A \sqrt {x}+y \relax (x )\right )} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-y[x]==-1/4*x+1/4*A*(x^(1/2)+5*A+3*A^2*x^(-1/2)),y[x],x,IncludeSingularSolutions -> True]
Not solved