Internal problem ID [10046]
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 1.3.3-2. Equations of
the form \(y y'=f_1(x) y+f_0(x)\)
Problem number: 52.
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-\frac {a \left (2 x -1\right ) y}{x^{\frac {5}{2}}}-\frac {a^{2} \left (x -1\right ) \left (3 x +1\right )}{2 x^{4}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 195
dsolve(y(x)*diff(y(x),x)-a*(2*x-1)*x^(-5/2)*y(x)=1/2*a^2*(x-1)*(3*x+1)*x^(-4),y(x), singsol=all)
\[ c_{1}-\frac {\left (\int _{}^{\frac {-18 x^{\frac {3}{2}} y \relax (x )+\left (-18 x -27\right ) a}{35 x \left (y \relax (x ) \sqrt {x}+a \right )}}\frac {\textit {\_a} \left (5 \textit {\_a} -9\right )^{\frac {1}{6}} \sqrt {7 \textit {\_a} +9}}{\left (35 \textit {\_a} +18\right )^{\frac {2}{3}} \left (-\frac {1225}{1458} \textit {\_a}^{3}+\frac {13}{6} \textit {\_a} +1\right )}d \textit {\_a} \right ) x \left (\frac {a}{x \left (-y \relax (x ) \sqrt {x}-a \right )}\right )^{\frac {2}{3}}-\frac {6 \sqrt {5}\, 21^{\frac {1}{6}} \sqrt {\frac {\left (x -1\right ) a +x^{\frac {3}{2}} y \relax (x )}{x \left (y \relax (x ) \sqrt {x}+a \right )}}\, 7^{\frac {2}{3}} \left (x +\frac {3}{2}\right ) 3^{\frac {5}{6}} \left (\frac {\left (-3 x -1\right ) a -3 x^{\frac {3}{2}} y \relax (x )}{x \left (y \relax (x ) \sqrt {x}+a \right )}\right )^{\frac {1}{6}}}{1225}}{\left (-\frac {a}{x \left (y \relax (x ) \sqrt {x}+a \right )}\right )^{\frac {2}{3}} x} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-a*(2*x-1)*x^(-5/2)*y[x]==1/2*a^2*(x-1)*(3*x+1)*x^(-4),y[x],x,IncludeSingularSolutions -> True]
Not solved