Internal problem ID [9960]
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]]
Solve \begin {gather*} \boxed {y y^{\prime }-y-6 x -\frac {A}{x^{4}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.001 (sec). Leaf size: 310
dsolve(y(x)*diff(y(x),x)-y(x)=6*x+A*x^(-4),y(x), singsol=all)
\[ c_{1}-\frac {x^{\frac {11}{2}} 625^{\frac {5}{6}} 243^{\frac {1}{6}} \left (-\frac {2}{x^{\frac {3}{2}} \left (10 x +5 y \relax (x )\right )}\right )^{\frac {4}{3}} A \hypergeom \left (\left [\frac {1}{6}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], -\frac {2 A}{3 x^{3} \left (2 x +y \relax (x )\right )^{2}}\right ) 16^{\frac {1}{6}} \left (-\frac {1}{x^{\frac {3}{2}} \left (2 x +y \relax (x )\right )}\right )^{\frac {5}{3}} 2^{\frac {2}{3}} \left (x +\frac {y \relax (x )}{2}\right )^{4} \left (\frac {12 x^{5}+12 y \relax (x ) x^{4}+3 x^{3} y \relax (x )^{2}+2 A}{x^{9} \left (2 x +y \relax (x )\right )^{6}}\right )^{\frac {1}{6}}-360 x^{\frac {5}{2}} \left (x^{5}+y \relax (x ) x^{4}+\frac {x^{3} y \relax (x )^{2}}{4}+\frac {A}{6}\right ) \left (-\frac {1}{x^{\frac {3}{2}} \left (2 x +y \relax (x )\right )}\right )^{\frac {5}{3}} \left (x +\frac {y \relax (x )}{2}\right )^{2} 2^{\frac {2}{3}}-10 \left (x +\frac {y \relax (x )}{2}\right ) 5^{\frac {2}{3}} \left (6 x^{5}+6 y \relax (x ) x^{4}+\frac {3 x^{3} y \relax (x )^{2}}{2}+A \right ) \left (-\frac {2}{5 x^{\frac {3}{2}} \left (2 x +y \relax (x )\right )}\right )^{\frac {2}{3}} y \relax (x )}{5 \left (\frac {12 x^{5}+12 y \relax (x ) x^{4}+3 x^{3} y \relax (x )^{2}+2 A}{x^{3} \left (2 x +y \relax (x )\right )^{2}}\right )^{\frac {1}{6}} \left (-\frac {2}{x^{\frac {3}{2}} \left (10 x +5 y \relax (x )\right )}\right )^{\frac {2}{3}} \left (-\frac {1}{x^{\frac {3}{2}} \left (2 x +y \relax (x )\right )}\right )^{\frac {5}{3}} \left (2 x +y \relax (x )\right )^{4} x^{\frac {11}{2}}} = 0 \]
✓ Solution by Mathematica
Time used: 1.206 (sec). Leaf size: 213
DSolve[y[x]*y'[x]-y[x]==6*x+A*x^(-4),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [c_1=\frac {i \left (-\frac {2 A+12 x^5+12 x^4 y(x)+3 x^3 y(x)^2}{A}\right )^{5/6} \left (-10\ 2^{5/6} x^5 \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {3}{2};-\frac {3 x^3 (2 x+y(x))^2}{2 A}\right )-5\ 2^{5/6} x^4 y(x) \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {3}{2};-\frac {3 x^3 (2 x+y(x))^2}{2 A}\right )+A \left (\frac {2 A+12 x^5+12 x^4 y(x)+3 x^3 y(x)^2}{A}\right )^{5/6}\right )}{2 \sqrt [3]{2} \sqrt {3} \sqrt {A} x^{5/2} \left (\frac {2 A+12 x^5+12 x^4 y(x)+3 x^3 y(x)^2}{A}\right )^{5/6}},y(x)\right ] \]