61.22.14 problem 14

Internal problem ID [12339]
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. subsection 1.3.1-2. Solvable equations and their solutions
Problem number : 14
Date solved : Tuesday, January 28, 2025 at 07:54:53 PM
CAS classification : [_rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} y^{\prime } y-y&=\frac {4}{9} x +2 A \,x^{2}+2 A^{2} x^{3} \end{align*}

✓ Solution by Maple

Time used: 0.002 (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)
 

\[ -\frac {27 \left (A y 3^{{1}/{4}} \left (x A +\frac {1}{3}\right ) {\left (\frac {A \left (3 A \,x^{2}+3 y+x \right ) \left (9 A^{2} x^{2}-9 y A +9 x A +2\right )}{\left (-9 y A +3 x A +1\right )^{2}}\right )}^{{1}/{4}}-\frac {\left (\int _{}^{\frac {\left (3 x A +1\right )^{2}}{-9 y A +3 x A +1}}\frac {\left (\textit {\_a}^{2}-1\right )^{{1}/{4}}}{\sqrt {\textit {\_a}}}d \textit {\_a} +c_{1} \right ) \left (\frac {1}{3}-3 y A +x A \right ) \sqrt {\frac {\left (3 x A +1\right )^{2}}{-9 y A +3 x A +1}}}{9}\right )}{\sqrt {\frac {\left (3 x A +1\right )^{2}}{-9 y A +3 x A +1}}\, \left (-9 y A +3 x A +1\right )} = 0 \]

✓ Solution by Mathematica

Time used: 2.147 (sec). Leaf size: 170

DSolve[y[x]*D[y[x],x]-y[x]==4/9*x+2*A*x^2+2*A^2*x^3,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\sqrt [4]{\frac {(-9 A y(x)+3 A x+1)^2}{(3 A x+1)^4}-1} \left (\frac {(-9 A y(x)+3 A x+1) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},\frac {(3 A x-9 A y(x)+1)^2}{(3 A x+1)^4}\right )}{2 \sqrt [4]{3} (3 A x+1) \sqrt {(3 A x+1)^2} \sqrt [4]{\frac {A \left (6 (3 A x+1) y(x)-27 A y(x)^2+x (3 A x+2) (3 A x+1)^2\right )}{(3 A x+1)^4}}}+\sqrt {(3 A x+1)^2}\right )+c_1=0,y(x)\right ] \]