Internal problem ID [10013]
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: 19.
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 (6 x -1\right ) y}{2 x}+\frac {a^{2} \left (x -1\right ) \left (4 x -1\right )}{2 x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 416
dsolve(y(x)*diff(y(x),x)+1/2*a*(6*x-1)*1/x*y(x)=-1/2*a^2*(x-1)*(4*x-1)*1/x,y(x), singsol=all)
\[ c_{1}+\frac {\sqrt {8}\, \left (\frac {i \left (i \sqrt {-2 x}\, \sqrt {2}\, a +4 a x +2 y \relax (x )-2 a \right ) \sqrt {-2 x}\, \sqrt {2}}{x a}\right )^{\frac {3}{2}} \left (-\frac {i \left (i \sqrt {-2 x}\, \sqrt {2}\, a +4 a x +2 y \relax (x )-2 a \right ) \sqrt {-2 x}\, \sqrt {2}\, \hypergeom \left (\left [\frac {1}{2}, \frac {3}{2}\right ], \left [\frac {7}{2}\right ], \frac {i \left (i \sqrt {-2 x}\, \sqrt {2}\, a +4 a x +2 y \relax (x )-2 a \right ) \sqrt {-2 x}\, \sqrt {2}}{8 x a}\right )}{32 x a}-\frac {15 \left (3 i \sqrt {-2 x}\, \sqrt {2}-4 i \sqrt {2}\, x +i \sqrt {2}-4 x -2\right ) \hypergeom \left (\left [-\frac {1}{2}, \frac {1}{2}\right ], \left [\frac {5}{2}\right ], \frac {i \left (i \sqrt {-2 x}\, \sqrt {2}\, a +4 a x +2 y \relax (x )-2 a \right ) \sqrt {-2 x}\, \sqrt {2}}{8 x a}\right )}{4 \left (2 i \sqrt {2}\, x +i \sqrt {2}-3 \sqrt {-2 x}+4 x -1\right ) \left (i \sqrt {2}+4\right )}\right )}{32 \left (\frac {3}{2}+\frac {\frac {9 i \sqrt {-2 x}\, \sqrt {2}}{2}-6 i \sqrt {2}\, x +\frac {3 i \sqrt {2}}{2}-6 x -3}{\left (2 i \sqrt {2}\, x +i \sqrt {2}-3 \sqrt {-2 x}+4 x -1\right ) \left (i \sqrt {2}+4\right )}\right ) \hypergeom \left (\left [-2, -1\right ], \left [-\frac {1}{2}\right ], \frac {i \left (i \sqrt {-2 x}\, \sqrt {2}\, a +4 a x +2 y \relax (x )-2 a \right ) \sqrt {-2 x}\, \sqrt {2}}{8 x a}\right )+\frac {16 i \left (i \sqrt {-2 x}\, \sqrt {2}\, a +4 a x +2 y \relax (x )-2 a \right ) \sqrt {-2 x}\, \sqrt {2}}{x a}} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]+1/2*a*(6*x-1)*1/x*y[x]==-1/2*a^2*(x-1)*(4*x-1)*1/x,y[x],x,IncludeSingularSolutions -> True]
Not solved