55.22.44 problem 44

Internal problem ID [13585]
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 : 44
Date solved : Sunday, October 12, 2025 at 03:58:06 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} y y^{\prime }-y&=A \,x^{2}-\frac {9}{625 A} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 186

ode:=y(x)*diff(y(x),x)-y(x) = A*x^2-9/625/A; 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {-\frac {125 \,2^{{5}/{6}} \left (\frac {-46875 A^{2} y^{2}+\left (37500 A^{2} x +4500 A \right ) y+31250 \left (A x -\frac {3}{25}\right ) \left (A x +\frac {3}{25}\right )^{2}}{\left (50 A x -125 y A +6\right )^{2}}\right )^{{1}/{6}} y A \sqrt {25 A x +3}}{2}+50 \left (A x -\frac {5 y A}{2}+\frac {3}{25}\right ) \left (\int _{}^{-\frac {2 \left (25 A x +3\right )^{{3}/{2}}}{125 y A -50 A x -6}}\frac {\left (\textit {\_a}^{2}-6\right )^{{1}/{6}}}{\textit {\_a}^{{1}/{3}}}d \textit {\_a} +c_1 \right ) \left (\frac {\left (25 A x +3\right )^{{3}/{2}}}{50 A x -125 y A +6}\right )^{{1}/{3}}}{\left (\frac {\left (25 A x +3\right )^{{3}/{2}}}{50 A x -125 y A +6}\right )^{{1}/{3}} \left (50 A x -125 y A +6\right )} = 0 \]

✓ Mathematica. Time used: 0.984 (sec). Leaf size: 198

ode=y[x]*D[y[x],x]-y[x]==A*x^2-9/625*A^(-1); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\frac {\sqrt [6]{\frac {46875 A^2 y(x)^2-1500 A (25 A x+3) y(x)-2 (25 A x-3) (25 A x+3)^2}{(25 A x+3)^3}} \left (\frac {(-125 A y(x)+50 A x+6) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{6},\frac {3}{2},\frac {3 (50 A x-125 A y(x)+6)^2}{2 (25 A x+3)^3}\right )}{\sqrt [3]{2} \sqrt {3} (25 A x+3)^{3/2} \sqrt [6]{\frac {-46875 A^2 y(x)^2+1500 A (25 A x+3) y(x)+2 (25 A x-3) (25 A x+3)^2}{(25 A x+3)^3}}}+\frac {\sqrt {25 A x+3}}{\sqrt {6}}\right )}{\sqrt [6]{2}}+c_1=0,y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
A = symbols("A") 
y = Function("y") 
ode = Eq(-A*x**2 + y(x)*Derivative(y(x), x) - y(x) + 9/(625*A),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out