61.22.45 problem 45

Internal problem ID [12370]
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 : 45
Date solved : Tuesday, January 28, 2025 at 07:57:16 PM
CAS classification : [_rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} y^{\prime } y-y&=-\frac {6}{25} x -A \,x^{2} \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 132

dsolve(y(x)*diff(y(x),x)-y(x)=-6/25*x-A*x^2,y(x), singsol=all)
 

\[ c_{1} +\frac {\left (2 x -5 y\right ) \left (\int _{}^{-\frac {10 \sqrt {-x A}\, x}{-2 x +5 y}}\frac {\left (\textit {\_a}^{2}-6\right )^{{1}/{6}}}{\textit {\_a}^{{1}/{3}}}d \textit {\_a} \right )-\frac {5 \,2^{{5}/{6}} \left (\frac {-50 A \,x^{3}-12 x^{2}+60 x y-75 y^{2}}{\left (-2 x +5 y\right )^{2}}\right )^{{1}/{6}} 5^{{2}/{3}} \sqrt {-x A}\, y}{2 \left (-\frac {\sqrt {-x A}\, x}{-2 x +5 y}\right )^{{1}/{3}}}}{2 x -5 y} = 0 \]

✓ Solution by Mathematica

Time used: 1.335 (sec). Leaf size: 162

DSolve[y[x]*D[y[x],x]-y[x]==-6/25*x-A*x^2,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [c_1=\frac {i \sqrt [6]{\frac {-2 x^2 (25 A x+6)+60 x y(x)-75 y(x)^2}{A x^3}} \left (25 A x^2-\frac {\sqrt [6]{2} \sqrt [3]{5} (2 x-5 y(x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{6},\frac {3}{2},-\frac {3 (2 x-5 y(x))^2}{50 A x^3}\right )}{\sqrt [6]{\frac {2 x^2 (25 A x+6)-60 x y(x)+75 y(x)^2}{A x^3}}}\right )}{5\ 2^{2/3} \sqrt {3} \sqrt [3]{5} \sqrt {A} x^{3/2}},y(x)\right ] \]