Internal problem ID [10025]
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: 31.
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 (x +1\right ) y}{2 x^{\frac {7}{4}}}-\frac {a^{2} \left (x -1\right ) \left (3 x +5\right )}{4 x^{\frac {5}{2}}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 192
dsolve(y(x)*diff(y(x),x)-1/2*a*(x+1)*x^(-7/4)*y(x)=1/4*a^2*(x-1)*(3*x+5)*x^(-5/2),y(x), singsol=all)
\[ c_{1}-\frac {\left (\int _{}^{\frac {-180 x^{\frac {3}{4}} y \relax (x )+\left (-180 x +1350\right ) a}{143 a x +143 x^{\frac {3}{4}} y \relax (x )}}\frac {\textit {\_a} \sqrt {11 \textit {\_a} -90}\, \left (13 \textit {\_a} +90\right )^{\frac {5}{6}}}{\left (143 \textit {\_a} +180\right )^{\frac {4}{3}} \left (-\frac {20449}{1458000} \textit {\_a}^{3}+\frac {49}{60} \textit {\_a} +1\right )}d \textit {\_a} \right ) x \left (\frac {a}{x^{\frac {3}{4}} y \relax (x )+a x}\right )^{\frac {4}{3}}-\frac {\sqrt {78}\, \left (\frac {\left (3 x +5\right ) a +3 x^{\frac {3}{4}} y \relax (x )}{x^{\frac {3}{4}} y \relax (x )+a x}\right )^{\frac {5}{6}} 3^{\frac {2}{3}} 30^{\frac {5}{6}} \left (x -\frac {15}{2}\right ) 11^{\frac {1}{6}} \sqrt {\frac {-x^{\frac {3}{4}} y \relax (x )-\left (x -1\right ) a}{x^{\frac {3}{4}} y \relax (x )+a x}}\, 1350^{\frac {2}{3}}}{4601025}}{\left (\frac {a}{x^{\frac {3}{4}} y \relax (x )+a x}\right )^{\frac {4}{3}} x} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-1/2*a*(x+1)*x^(-7/4)*y[x]==1/4*a^2*(x-1)*(3*x+5)*x^(-5/2),y[x],x,IncludeSingularSolutions -> True]
Not solved