Internal problem ID [9956]
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: 54.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_Abel, `2nd type`, `class B`]]
\[ \boxed {y^{\prime } y-y-6 x -\frac {A}{x^{4}}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 308
dsolve(y(x)*diff(y(x),x)-y(x)=6*x+A*x^(-4),y(x), singsol=all)
\[ c_{1} +\frac {2 \left (-\frac {x^{\frac {11}{2}} 625^{\frac {5}{6}} 2^{\frac {2}{3}} 16^{\frac {1}{6}} A \operatorname {hypergeom}\left (\left [\frac {1}{6}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], -\frac {2 A}{3 x^{3} \left (y \left (x \right )+2 x \right )^{2}}\right ) 243^{\frac {1}{6}} \left (-\frac {1}{x^{\frac {3}{2}} \left (y \left (x \right )+2 x \right )}\right )^{\frac {5}{3}} \left (\frac {y \left (x \right )}{2}+x \right )^{3} \left (\frac {12 x^{5}+12 x^{4} y \left (x \right )+3 x^{3} y \left (x \right )^{2}+2 A}{x^{9} \left (y \left (x \right )+2 x \right )^{6}}\right )^{\frac {1}{6}} \left (-\frac {2}{x^{\frac {3}{2}} \left (10 x +5 y \left (x \right )\right )}\right )^{\frac {4}{3}}}{10}+36 x^{\frac {5}{2}} \left (-\frac {1}{x^{\frac {3}{2}} \left (y \left (x \right )+2 x \right )}\right )^{\frac {5}{3}} \left (\frac {y \left (x \right )}{2}+x \right ) \left (x^{5}+x^{4} y \left (x \right )+\frac {x^{3} y \left (x \right )^{2}}{4}+\frac {A}{6}\right ) 2^{\frac {2}{3}}+y \left (x \right ) 5^{\frac {2}{3}} \left (-\frac {2}{5 x^{\frac {3}{2}} \left (y \left (x \right )+2 x \right )}\right )^{\frac {2}{3}} \left (6 x^{5}+6 x^{4} y \left (x \right )+\frac {3 x^{3} y \left (x \right )^{2}}{2}+A \right )\right ) \left (\frac {y \left (x \right )}{2}+x \right )}{\left (-\frac {2}{x^{\frac {3}{2}} \left (10 x +5 y \left (x \right )\right )}\right )^{\frac {2}{3}} \left (-\frac {1}{x^{\frac {3}{2}} \left (y \left (x \right )+2 x \right )}\right )^{\frac {5}{3}} \left (\frac {12 x^{5}+12 x^{4} y \left (x \right )+3 x^{3} y \left (x \right )^{2}+2 A}{x^{3} \left (y \left (x \right )+2 x \right )^{2}}\right )^{\frac {1}{6}} \left (y \left (x \right )+2 x \right )^{4} x^{\frac {11}{2}}} = 0 \]
✓ Solution by Mathematica
Time used: 1.189 (sec). Leaf size: 213
DSolve[y[x]*y'[x]-y[x]==6*x+A*x^(-4),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [c_1=\frac {i \left (-\frac {2 A+12 x^5+12 x^4 y(x)+3 x^3 y(x)^2}{A}\right )^{5/6} \left (-10\ 2^{5/6} x^5 \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {3}{2},-\frac {3 x^3 (2 x+y(x))^2}{2 A}\right )-5\ 2^{5/6} x^4 y(x) \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {3}{2},-\frac {3 x^3 (2 x+y(x))^2}{2 A}\right )+A \left (\frac {2 A+12 x^5+12 x^4 y(x)+3 x^3 y(x)^2}{A}\right )^{5/6}\right )}{2 \sqrt [3]{2} \sqrt {3} \sqrt {A} x^{5/2} \left (\frac {2 A+12 x^5+12 x^4 y(x)+3 x^3 y(x)^2}{A}\right )^{5/6}},y(x)\right ] \]