61.24.19 problem 19

Internal problem ID [12432]
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.
Problem number : 19
Date solved : Tuesday, January 28, 2025 at 07:58:49 PM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} 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} \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 340

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)
 

\[ c_{1} +\frac {\sqrt {2}\, \left (\frac {i \left (i \sqrt {-x}\, a +2 a x +y-a \right ) \sqrt {-x}}{x a}\right )^{{3}/{2}} \left (-\frac {i \left (i \sqrt {-x}\, a +2 a x +y-a \right ) \sqrt {-x}\, \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {3}{2}\right ], \left [\frac {7}{2}\right ], \frac {i \left (i \sqrt {-x}\, a +2 a x +y-a \right ) \sqrt {-x}}{2 x a}\right )}{8 x a}+\frac {5 \left (-4 i \sqrt {2}\, x +6 i \sqrt {-x}+i \sqrt {2}-4 x -2\right ) \operatorname {hypergeom}\left (\left [-\frac {1}{2}, \frac {1}{2}\right ], \left [\frac {5}{2}\right ], \frac {i \left (i \sqrt {-x}\, a +2 a x +y-a \right ) \sqrt {-x}}{2 x a}\right )}{4 \left (-4 i \sqrt {2}\, x +4 \sqrt {2}\, \sqrt {-x}+2 i \sqrt {-x}-i \sqrt {2}-4 x +2\right )}\right )}{2 \left (\frac {3}{2}-\frac {-4 i \sqrt {2}\, x +6 i \sqrt {-x}+i \sqrt {2}-4 x -2}{2 \left (-4 i \sqrt {2}\, x +4 \sqrt {2}\, \sqrt {-x}+2 i \sqrt {-x}-i \sqrt {2}-4 x +2\right )}\right ) \operatorname {hypergeom}\left (\left [-2, -1\right ], \left [-\frac {1}{2}\right ], \frac {i \left (i \sqrt {-x}\, a +2 a x +y-a \right ) \sqrt {-x}}{2 x a}\right )+\frac {4 i \left (i \sqrt {-x}\, a +2 a x +y-a \right ) \sqrt {-x}}{x a}} = 0 \]

✗ Solution by Mathematica

Time used: 0.000 (sec). Leaf size: 0

DSolve[y[x]*D[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],x,IncludeSingularSolutions -> True]
 

Not solved