Internal problem ID [9964]
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: 58.
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 {12 x}{49}-\frac {2 A \left (\sqrt {x}+166 A +\frac {55 A^{2}}{\sqrt {x}}\right )}{49}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 692
dsolve(y(x)*diff(y(x),x)-y(x)=-12/49*x+2/49*A*(x^(1/2)+166*A+55*A^2*x^(-1/2)),y(x), singsol=all)
\[ c_{1}+\frac {-\frac {2 i \left (5 A +\sqrt {x}\right ) \sqrt {-7 A \left (5 A +\sqrt {x}\right )}\, \sqrt {6}}{3 \left (-10 i A \sqrt {6}\, \sqrt {-7 A \left (5 A +\sqrt {x}\right )}-2 i \sqrt {-7 A \left (5 A +\sqrt {x}\right )}\, \sqrt {6}\, \sqrt {x}+120 A^{2}-36 A \sqrt {x}-12 x +21 y \relax (x )\right )}+\frac {2 \left (5 A +\sqrt {x}+\sqrt {-7 A \left (5 A +\sqrt {x}\right )}\right ) \hypergeom \left (\left [-1, -\frac {1}{6}\right ], \left [\frac {2}{3}\right ], -\frac {4 i \left (5 A +\sqrt {x}\right ) \sqrt {-7 A \left (5 A +\sqrt {x}\right )}\, \sqrt {6}}{-10 i A \sqrt {6}\, \sqrt {-7 A \left (5 A +\sqrt {x}\right )}-2 i \sqrt {-7 A \left (5 A +\sqrt {x}\right )}\, \sqrt {6}\, \sqrt {x}+120 A^{2}-36 A \sqrt {x}-12 x +21 y \relax (x )}\right )}{3 \left (7 i \sqrt {6}\, A -i \sqrt {6}\, \sqrt {-7 A \left (5 A +\sqrt {x}\right )}-15 A -3 \sqrt {-7 A \left (5 A +\sqrt {x}\right )}-3 \sqrt {x}\right )}}{\left (-\frac {4 i \left (5 A +\sqrt {x}\right ) \sqrt {-7 A \left (5 A +\sqrt {x}\right )}\, \sqrt {6}}{-10 i A \sqrt {6}\, \sqrt {-7 A \left (5 A +\sqrt {x}\right )}-2 i \sqrt {-7 A \left (5 A +\sqrt {x}\right )}\, \sqrt {6}\, \sqrt {x}+120 A^{2}-36 A \sqrt {x}-12 x +21 y \relax (x )}\right )^{\frac {1}{3}} \left (-\frac {4 \left (-\frac {1}{3}-\frac {5 A +\sqrt {x}+\sqrt {-7 A \left (5 A +\sqrt {x}\right )}}{7 i \sqrt {6}\, A -i \sqrt {6}\, \sqrt {-7 A \left (5 A +\sqrt {x}\right )}-15 A -3 \sqrt {-7 A \left (5 A +\sqrt {x}\right )}-3 \sqrt {x}}\right ) \hypergeom \left (\left [-\frac {2}{3}, \frac {1}{6}\right ], \left [\frac {4}{3}\right ], -\frac {4 i \left (5 A +\sqrt {x}\right ) \sqrt {-7 A \left (5 A +\sqrt {x}\right )}\, \sqrt {6}}{-10 i A \sqrt {6}\, \sqrt {-7 A \left (5 A +\sqrt {x}\right )}-2 i \sqrt {-7 A \left (5 A +\sqrt {x}\right )}\, \sqrt {6}\, \sqrt {x}+120 A^{2}-36 A \sqrt {x}-12 x +21 y \relax (x )}\right )}{3}+\frac {4 i \left (5 A +\sqrt {x}\right ) \sqrt {-7 A \left (5 A +\sqrt {x}\right )}\, \sqrt {6}\, \hypergeom \left (\left [\frac {1}{3}, \frac {7}{6}\right ], \left [\frac {7}{3}\right ], -\frac {4 i \left (5 A +\sqrt {x}\right ) \sqrt {-7 A \left (5 A +\sqrt {x}\right )}\, \sqrt {6}}{-10 i A \sqrt {6}\, \sqrt {-7 A \left (5 A +\sqrt {x}\right )}-2 i \sqrt {-7 A \left (5 A +\sqrt {x}\right )}\, \sqrt {6}\, \sqrt {x}+120 A^{2}-36 A \sqrt {x}-12 x +21 y \relax (x )}\right )}{9 \left (-10 i A \sqrt {6}\, \sqrt {-7 A \left (5 A +\sqrt {x}\right )}-2 i \sqrt {-7 A \left (5 A +\sqrt {x}\right )}\, \sqrt {6}\, \sqrt {x}+120 A^{2}-36 A \sqrt {x}-12 x +21 y \relax (x )\right )}\right )} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-y[x]==-12/49*x+2/49*A*(x^(1/2)+166*A+55*A^2*x^(-1/2)),y[x],x,IncludeSingularSolutions -> True]
Not solved