Internal problem ID [9958]
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: 52.
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 {A \left (5 \sqrt {x}+262 A +\frac {65 A^{2}}{\sqrt {x}}\right )}{49}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 703
dsolve(y(x)*diff(y(x),x)-y(x)=-12/49*x+1/49*A*(5*x^(1/2)+262*A+65*A^2*x^(-1/2)),y(x), singsol=all)
\[ c_{1}+\frac {6 i \left (\left (\left (\left (\frac {5 i A \sqrt {3}}{3}+3 A \right ) \sqrt {x}+\left (-\frac {25}{6} i A^{2}-\frac {1}{6} i x \right ) \sqrt {3}+10 A^{2}-x +\frac {7 y \relax (x )}{4}\right ) \sqrt {-35 A^{2}+7 A \sqrt {x}}+\frac {7 i \sqrt {3}\, x A}{6}+\left (5 A^{2}-\frac {7 y \relax (x )}{4}-\frac {35 i A^{2} \sqrt {3}}{3}\right ) \sqrt {x}+\frac {175 i A^{3} \sqrt {3}}{6}+50 A^{3}+\left (-8 x +\frac {35 y \relax (x )}{4}\right ) A +x^{\frac {3}{2}}\right ) \hypergeom \left (\left [-\frac {4}{3}, -\frac {1}{6}\right ], \left [\frac {2}{3}\right ], \frac {4 i \left (5 A -\sqrt {x}\right ) \sqrt {-35 A^{2}+7 A \sqrt {x}}\, \sqrt {3}}{10 i \left (A -\frac {\sqrt {x}}{5}\right ) \sqrt {3}\, \sqrt {-35 A^{2}+7 A \sqrt {x}}-120 A^{2}-36 A \sqrt {x}+12 x -21 y \relax (x )}\right )-\frac {175 \hypergeom \left (\left [-\frac {1}{3}, \frac {5}{6}\right ], \left [\frac {5}{3}\right ], \frac {4 i \left (5 A -\sqrt {x}\right ) \sqrt {-35 A^{2}+7 A \sqrt {x}}\, \sqrt {3}}{10 i \left (A -\frac {\sqrt {x}}{5}\right ) \sqrt {3}\, \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 {2 i A \sqrt {3}}{35}+\frac {3 A}{50}\right ) \sqrt {x}+\left (-\frac {1}{7} i A^{2}-\frac {1}{175} i x \right ) \sqrt {3}-\frac {3 A^{2}}{10}\right ) \sqrt {-35 A^{2}+7 A \sqrt {x}}+\left (i \sqrt {3}-\frac {3}{2}\right ) \left (A -\frac {\sqrt {x}}{5}\right )^{2} A \right )}{3}\right ) \sqrt {3}\, 4^{\frac {2}{3}}}{\left (\frac {i \left (5 A -\sqrt {x}\right ) \sqrt {-35 A^{2}+7 A \sqrt {x}}\, \sqrt {3}}{10 i \left (A -\frac {\sqrt {x}}{5}\right ) \sqrt {3}\, \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 (-84 i A \sqrt {3}+216 A \right ) \sqrt {x}+420 i A^{2} \sqrt {3}+720 A^{2}-72 x +126 y \relax (x )\right ) \sqrt {-35 A^{2}+7 A \sqrt {x}}+2100 \left (\frac {i \sqrt {3}\, x}{25}+\left (-\frac {2 i A \sqrt {3}}{5}-\frac {18 A}{25}\right ) \sqrt {x}+i A^{2} \sqrt {3}-\frac {12 A^{2}}{5}+\frac {6 x}{25}-\frac {21 y \relax (x )}{50}\right ) A \right ) \hypergeom \left (\left [-1, \frac {1}{6}\right ], \left [\frac {4}{3}\right ], \frac {4 i \left (5 A -\sqrt {x}\right ) \sqrt {-35 A^{2}+7 A \sqrt {x}}\, \sqrt {3}}{10 i \left (A -\frac {\sqrt {x}}{5}\right ) \sqrt {3}\, \sqrt {-35 A^{2}+7 A \sqrt {x}}-120 A^{2}-36 A \sqrt {x}+12 x -21 y \relax (x )}\right )+\left (\left (63 i A \sqrt {3}-180 A \right ) \sqrt {x}-315 i A^{2} \sqrt {3}+450 A^{2}+18 x \right ) \sqrt {-35 A^{2}+7 A \sqrt {x}}-1575 \left (A -\frac {\sqrt {x}}{5}\right )^{2} A \left (i \sqrt {3}+2\right )\right )} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-y[x]==-28/121*x+2/121*A*(5*x^(1/2)+262*A+65*A^2*x^(-1/2)),y[x],x,IncludeSingularSolutions -> True]
Not solved