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 y^{\prime }-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: 0.002 (sec). Leaf size: 693
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 {3 i \sqrt {6}\, \left (\left (\left (\left (\frac {5 i A \sqrt {6}}{3}+3 A \right ) \sqrt {x}+\left (\frac {1}{6} i x +\frac {25}{6} i A^{2}\right ) \sqrt {6}-10 A^{2}+x -\frac {7 y \relax (x )}{4}\right ) \sqrt {-35 A^{2}-7 A \sqrt {x}}-\frac {7 i \sqrt {6}\, x A}{6}+\left (5 A^{2}-\frac {7 y \relax (x )}{4}-\frac {35 i A^{2} \sqrt {6}}{3}\right ) \sqrt {x}-\frac {175 i A^{3} \sqrt {6}}{6}-50 A^{3}+\left (8 x -\frac {35 y \relax (x )}{4}\right ) A +x^{\frac {3}{2}}\right ) \hypergeom \left (\left [-1, -\frac {1}{6}\right ], \left [\frac {2}{3}\right ], \frac {4 i \left (5 A +\sqrt {x}\right ) \sqrt {6}\, \sqrt {-35 A^{2}-7 A \sqrt {x}}}{10 i \sqrt {6}\, \left (A +\frac {\sqrt {x}}{5}\right ) \sqrt {-35 A^{2}-7 A \sqrt {x}}-120 A^{2}+36 A \sqrt {x}+12 x -21 y \relax (x )}\right )+\left (\left (-\frac {7 A}{2}-\frac {5 i A \sqrt {6}}{2}\right ) \sqrt {x}+\left (-\frac {1}{4} i x -\frac {25}{4} i A^{2}\right ) \sqrt {6}-\frac {35 A^{2}}{2}\right ) \sqrt {-35 A^{2}-7 A \sqrt {x}}+\frac {175 A \left (A +\frac {\sqrt {x}}{5}\right )^{2} \left (i \sqrt {6}-2\right )}{4}\right ) 4^{\frac {2}{3}}}{\left (\frac {i \left (5 A +\sqrt {x}\right ) \sqrt {6}\, \sqrt {-35 A^{2}-7 A \sqrt {x}}}{10 i \sqrt {6}\, \left (A +\frac {\sqrt {x}}{5}\right ) \sqrt {-35 A^{2}-7 A \sqrt {x}}-120 A^{2}+36 A \sqrt {x}+12 x -21 y \relax (x )}\right )^{\frac {1}{3}} \left (\left (\left (\left (-56 i A \sqrt {6}+144 A \right ) \sqrt {x}-280 i A^{2} \sqrt {6}-480 A^{2}+48 x -84 y \relax (x )\right ) \sqrt {-35 A^{2}-7 A \sqrt {x}}-1400 A \left (\frac {i \sqrt {6}\, x}{25}+\left (\frac {2 i A \sqrt {6}}{5}+\frac {18 A}{25}\right ) \sqrt {x}+i A^{2} \sqrt {6}-\frac {12 A^{2}}{5}+\frac {6 x}{25}-\frac {21 y \relax (x )}{50}\right )\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 {6}\, \sqrt {-35 A^{2}-7 A \sqrt {x}}}{10 i \sqrt {6}\, \left (A +\frac {\sqrt {x}}{5}\right ) \sqrt {-35 A^{2}-7 A \sqrt {x}}-120 A^{2}+36 A \sqrt {x}+12 x -21 y \relax (x )}\right )+700 \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 {6}\, \sqrt {-35 A^{2}-7 A \sqrt {x}}}{10 i \sqrt {6}\, \left (A +\frac {\sqrt {x}}{5}\right ) \sqrt {-35 A^{2}-7 A \sqrt {x}}-120 A^{2}+36 A \sqrt {x}+12 x -21 y \relax (x )}\right ) \left (\left (\left (\frac {i A \sqrt {6}}{25}-\frac {6 A}{35}\right ) \sqrt {x}+\frac {i A^{2} \sqrt {6}}{5}-\frac {3 A^{2}}{7}-\frac {3 x}{175}\right ) \sqrt {-35 A^{2}-7 A \sqrt {x}}+\left (i \sqrt {6}+3\right ) A \left (A +\frac {\sqrt {x}}{5}\right )^{2}\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