Internal problem ID [9954]
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`]]
\[ \boxed {y^{\prime } y-y+\frac {12 x}{49}-\frac {A \left (5 \sqrt {x}+262 A +\frac {65 A^{2}}{\sqrt {x}}\right )}{49}=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 697
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 {2 i \sqrt {3}\, 4^{\frac {2}{3}} \left (\left (\left (A \left (3+\frac {5 i \sqrt {3}}{3}\right ) \sqrt {x}+\frac {i \left (-25 A^{2}-x \right ) \sqrt {3}}{6}-x +\frac {7 y \left (x \right )}{4}+10 A^{2}\right ) \sqrt {-35 A^{2}+7 A \sqrt {x}}+\frac {7 i A x \sqrt {3}}{6}+\left (-\frac {35 i A^{2} \sqrt {3}}{3}+5 A^{2}-\frac {7 y \left (x \right )}{4}\right ) \sqrt {x}+\frac {175 i A^{3} \sqrt {3}}{6}+50 A^{3}+\left (-8 x +\frac {35 y \left (x \right )}{4}\right ) A +x^{\frac {3}{2}}\right ) \operatorname {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 {3}\, \sqrt {-35 A^{2}+7 A \sqrt {x}}}{10 i \sqrt {3}\, \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 {175 \operatorname {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 {3}\, \sqrt {-35 A^{2}+7 A \sqrt {x}}}{10 i \sqrt {3}\, \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 {3}{50}+\frac {2 i \sqrt {3}}{35}\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}}+A \left (A -\frac {\sqrt {x}}{5}\right )^{2} \left (i \sqrt {3}-\frac {3}{2}\right )\right )}{3}\right )}{{\left (\frac {i \left (5 A -\sqrt {x}\right ) \sqrt {3}\, \sqrt {-35 A^{2}+7 A \sqrt {x}}}{10 i \sqrt {3}\, \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 (2 \left (\left (2 A \left (18-7 i \sqrt {3}\right ) \sqrt {x}+70 i A^{2} \sqrt {3}+120 A^{2}-12 x +21 y \left (x \right )\right ) \sqrt {-35 A^{2}+7 A \sqrt {x}}+350 A \left (\frac {i \sqrt {3}\, x}{25}+2 A \left (-\frac {9}{25}-\frac {i \sqrt {3}}{5}\right ) \sqrt {x}+i A^{2} \sqrt {3}-\frac {12 A^{2}}{5}+\frac {6 x}{25}-\frac {21 y \left (x \right )}{50}\right )\right ) \operatorname {hypergeom}\left (\left [-1, \frac {1}{6}\right ], \left [\frac {4}{3}\right ], \frac {4 i \left (5 A -\sqrt {x}\right ) \sqrt {3}\, \sqrt {-35 A^{2}+7 A \sqrt {x}}}{10 i \sqrt {3}\, \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 )+3 \left (A \left (-20+7 i \sqrt {3}\right ) \sqrt {x}-35 i A^{2} \sqrt {3}+50 A^{2}+2 x \right ) \sqrt {-35 A^{2}+7 A \sqrt {x}}-525 A \left (A -\frac {\sqrt {x}}{5}\right )^{2} \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