Internal problem ID [10717]
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`]]
\[ \boxed {y y^{\prime }-y=-\frac {12 x}{49}+\frac {2 A \left (\sqrt {x}+166 A +\frac {55 A^{2}}{\sqrt {x}}\right )}{49}} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 683
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}\, 4^{\frac {2}{3}} \left (\left (\left (A \left (3+\frac {5 i \sqrt {6}}{3}\right ) \sqrt {x}+\frac {i \left (25 A^{2}+x \right ) \sqrt {6}}{6}-\frac {7 y \left (x \right )}{4}-10 A^{2}+x \right ) \sqrt {-35 A^{2}-7 A \sqrt {x}}-\frac {7 i A x \sqrt {6}}{6}+\left (-\frac {35 i A^{2} \sqrt {6}}{3}+5 A^{2}-\frac {7 y \left (x \right )}{4}\right ) \sqrt {x}-\frac {175 i A^{3} \sqrt {6}}{6}-50 A^{3}+\left (-\frac {35 y \left (x \right )}{4}+8 x \right ) A +x^{\frac {3}{2}}\right ) \operatorname {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 \left (x \right )}\right )+\left (A \left (-\frac {7}{2}-\frac {5 i \sqrt {6}}{2}\right ) \sqrt {x}+\frac {i \left (-25 A^{2}-x \right ) \sqrt {6}}{4}-\frac {35 A^{2}}{2}\right ) \sqrt {-35 A^{2}-7 A \sqrt {x}}+\frac {175 \left (i \sqrt {6}-2\right ) A \left (A +\frac {\sqrt {x}}{5}\right )^{2}}{4}\right )}{4 {\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 \left (x \right )}\right )}^{\frac {1}{3}} \left (\left (\left (2 A \left (18-7 i \sqrt {6}\right ) \sqrt {x}-70 i A^{2} \sqrt {6}-120 A^{2}+12 x -21 y \left (x \right )\right ) \sqrt {-35 A^{2}-7 A \sqrt {x}}-350 \left (\frac {i x \sqrt {6}}{25}+2 A \left (\frac {9}{25}+\frac {i \sqrt {6}}{5}\right ) \sqrt {x}+i A^{2} \sqrt {6}-\frac {12 A^{2}}{5}+\frac {6 x}{25}-\frac {21 y \left (x \right )}{50}\right ) A \right ) \operatorname {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 \left (x \right )}\right )+175 \operatorname {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 \left (x \right )}\right ) \left (\left (A \left (-\frac {6}{35}+\frac {i \sqrt {6}}{25}\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