Internal problem ID [9920]
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: 14.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_Abel, 2nd type, class A]]
Solve \begin {gather*} \boxed {y^{\prime } y-y-\frac {4 x}{9}-2 A \,x^{2}-2 A^{2} x^{3}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 177
dsolve(y(x)*diff(y(x),x)-y(x)=4/9*x+2*A*x^2+2*A^2*x^3,y(x), singsol=all)
\[ c_{1}-\frac {27 \left (\sqrt {3}\, y \relax (x ) A \left (x A +\frac {1}{3}\right ) \left (\frac {A \left (\frac {2}{9}+A^{2} x^{2}+\left (x -y \relax (x )\right ) A \right ) \left (A \,x^{2}+\frac {x}{3}+y \relax (x )\right )}{\left (\frac {1}{3}+\left (x -3 y \relax (x )\right ) A \right )^{2}}\right )^{\frac {1}{4}}-\frac {\left (\int _{}^{\frac {\left (3 x A +1\right )^{2}}{1+\left (3 x -9 y \relax (x )\right ) A}}\frac {\left (\textit {\_a}^{2}-1\right )^{\frac {1}{4}}}{\sqrt {\textit {\_a}}}d \textit {\_a} \right ) \sqrt {\frac {\left (3 x A +1\right )^{2}}{1+\left (3 x -9 y \relax (x )\right ) A}}\, \left (\frac {1}{3}+\left (x -3 y \relax (x )\right ) A \right )}{9}\right )}{\sqrt {\frac {\left (3 x A +1\right )^{2}}{1+\left (3 x -9 y \relax (x )\right ) A}}\, \left (1+\left (3 x -9 y \relax (x )\right ) A \right )} = 0 \]
✓ Solution by Mathematica
Time used: 1.929 (sec). Leaf size: 170
DSolve[y[x]*y'[x]-y[x]==4/9*x+2*A*x^2+2*A^2*x^3,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\sqrt [4]{\frac {(-9 A y(x)+3 A x+1)^2}{(3 A x+1)^4}-1} \left (\frac {(-9 A y(x)+3 A x+1) \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {3}{2};\frac {(3 A x-9 A y(x)+1)^2}{(3 A x+1)^4}\right )}{2 \sqrt [4]{3} (3 A x+1) \sqrt {(3 A x+1)^2} \sqrt [4]{\frac {A \left (6 (3 A x+1) y(x)-27 A y(x)^2+x (3 A x+2) (3 A x+1)^2\right )}{(3 A x+1)^4}}}+\sqrt {(3 A x+1)^2}\right )+c_1=0,y(x)\right ] \]