Internal problem ID [9948]
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: 46.
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: 0.0 (sec). Leaf size: 196
dsolve(y(x)*diff(y(x),x)-y(x)=6/25*x-A*x^2,y(x), singsol=all)
\[ c_{1} +\frac {-125 \,5^{\frac {1}{3}} 2^{\frac {5}{6}} \left (\frac {-1250 A^{3} x^{3}+\left (600 x^{2}+1500 x y \left (x \right )-1875 y \left (x \right )^{2}\right ) A^{2}+\left (-72 x -360 y \left (x \right )\right ) A}{\left (50 x A -125 A y \left (x \right )-12\right )^{2}}\right )^{\frac {1}{6}} A y \left (x \right ) \sqrt {-25 x A +6}+100 \left (-\frac {6}{25}+\left (x -\frac {5 y \left (x \right )}{2}\right ) A \right ) \left (\int _{}^{-\frac {2 \left (-25 x A +6\right )^{\frac {3}{2}}}{-12+\left (50 x -125 y \left (x \right )\right ) A}}\frac {\left (\textit {\_a}^{2}-6\right )^{\frac {1}{6}}}{\textit {\_a}^{\frac {1}{3}}}d \textit {\_a} \right ) \left (\frac {\left (-25 x A +6\right )^{\frac {3}{2}}}{-50 x A +125 A y \left (x \right )+12}\right )^{\frac {1}{3}}}{\left (-\frac {\left (-25 x A +6\right )^{\frac {3}{2}}}{-12+\left (50 x -125 y \left (x \right )\right ) A}\right )^{\frac {1}{3}} \left (-24+\left (100 x -250 y \left (x \right )\right ) A \right )} = 0 \]
✓ Solution by Mathematica
Time used: 2.015 (sec). Leaf size: 189
DSolve[y[x]*y'[x]-y[x]==6/25*x-A*x^2,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {\sqrt [3]{5} \sqrt [6]{-\frac {A \left (1875 A y(x)^2-60 (25 A x-6) y(x)+2 x (6-25 A x)^2\right )}{(25 A x-6)^3}} \left (\frac {(-125 A y(x)+50 A x-12) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{6},\frac {3}{2},-\frac {3 (-50 A x+125 A y(x)+12)^2}{2 (25 A x-6)^3}\right )}{\sqrt [3]{10} \sqrt {18-75 A x} (25 A x-6) \sqrt [6]{\frac {A \left (1875 A y(x)^2-60 (25 A x-6) y(x)+2 x (6-25 A x)^2\right )}{(25 A x-6)^3}}}+\sqrt {1-\frac {25 A x}{6}}\right )}{\sqrt [6]{2}}+c_1=0,y(x)\right ] \]