Internal problem ID [9947]
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: 45.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_Abel, `2nd type`, `class A`]]
\[ \boxed {y^{\prime } y-y+\frac {6 x}{25}+A \,x^{2}=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 160
dsolve(y(x)*diff(y(x),x)-y(x)=-6/25*x-A*x^2,y(x), singsol=all)
\[ c_{1} +\frac {\left (2 x -5 y \left (x \right )\right ) \left (\int _{}^{\frac {10 \sqrt {-x A}\, x}{2 x -5 y \left (x \right )}}\frac {\left (\textit {\_a}^{2}-6\right )^{\frac {1}{6}}}{\textit {\_a}^{\frac {1}{3}}}d \textit {\_a} \right )-\frac {5 \,2^{\frac {1}{6}} \left (-\frac {50 x^{3} A}{\left (2 x -5 y \left (x \right )\right )^{2}}-\frac {12 x^{2}}{\left (2 x -5 y \left (x \right )\right )^{2}}+\frac {60 y \left (x \right ) x}{\left (2 x -5 y \left (x \right )\right )^{2}}-\frac {75 y \left (x \right )^{2}}{\left (2 x -5 y \left (x \right )\right )^{2}}\right )^{\frac {1}{6}} 10^{\frac {2}{3}} \sqrt {-x A}\, y \left (x \right )}{2 \left (\frac {\sqrt {-x A}\, x}{2 x -5 y \left (x \right )}\right )^{\frac {1}{3}}}}{2 x -5 y \left (x \right )} = 0 \]
✓ Solution by Mathematica
Time used: 1.313 (sec). Leaf size: 162
DSolve[y[x]*y'[x]-y[x]==-6/25*x-A*x^2,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [c_1=\frac {i \sqrt [6]{\frac {-2 x^2 (25 A x+6)+60 x y(x)-75 y(x)^2}{A x^3}} \left (25 A x^2-\frac {\sqrt [6]{2} \sqrt [3]{5} (2 x-5 y(x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{6},\frac {3}{2},-\frac {3 (2 x-5 y(x))^2}{50 A x^3}\right )}{\sqrt [6]{\frac {2 x^2 (25 A x+6)-60 x y(x)+75 y(x)^2}{A x^3}}}\right )}{5\ 2^{2/3} \sqrt {3} \sqrt [3]{5} \sqrt {A} x^{3/2}},y(x)\right ] \]