55.26.20 problem 20

Internal problem ID [13767]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.4. Equations Containing Polynomial Functions of y. subsection 1.4.1-2 Abel equations of the first kind.
Problem number : 20
Date solved : Thursday, October 02, 2025 at 08:01:28 AM
CAS classification : [_Abel]

\begin{align*} y^{\prime }&=-\frac {\left (a x -\frac {6}{25}\right )^{{34}/{9}} y^{3}}{x^{{16}/{9}}}+\frac {\frac {2 a x}{3}-\frac {4}{675}}{x^{{11}/{18}} \left (a x -\frac {6}{25}\right )^{{61}/{18}}} \end{align*}

✗ Maple

ode:=diff(y(x),x) = -1/x^(16/9)*(a*x-6/25)^(34/9)*y(x)^3+2/27*(9*a*x-2/25)/x^(11/18)/(a*x-6/25)^(61/18); 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 4.947 (sec). Leaf size: 487

ode=D[y[x],x]==-x^(-16/9)*(a*x-6/25)^(34/9)*y[x]^3+2/27*(9*a*x-2/25)*x^(-11/18)*(a*x-6/25)^(-61/18); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [c_1=\sqrt [6]{\left (\frac {1}{\frac {\sqrt {\frac {2}{3}} \sqrt {a} \sqrt [9]{x} \left (25 a x (25 a x-6)^{7/18}-6 (25 a x-6)^{7/18}\right ) y(x)}{25\ 5^{7/9}}+\frac {25 \sqrt {\frac {2}{3}} \sqrt {a} \sqrt {x}}{3 (25 a x-6)}}+\frac {\sqrt {6}}{5 \sqrt {a} \sqrt {x}}\right )^2-1} \left (-\frac {\left (\frac {1}{\frac {\sqrt {\frac {2}{3}} \sqrt {a} \sqrt [9]{x} \left (25 a x (25 a x-6)^{7/18}-6 (25 a x-6)^{7/18}\right ) y(x)}{25\ 5^{7/9}}+\frac {25 \sqrt {\frac {2}{3}} \sqrt {a} \sqrt {x}}{3 (25 a x-6)}}+\frac {\sqrt {6}}{5 \sqrt {a} \sqrt {x}}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{6},\frac {3}{2},\left (\frac {1}{\frac {\sqrt {\frac {2}{3}} \sqrt {a} \sqrt [9]{x} \left (25 a x (25 a x-6)^{7/18}-6 (25 a x-6)^{7/18}\right ) y(x)}{25\ 5^{7/9}}+\frac {25 \sqrt {\frac {2}{3}} \sqrt {a} \sqrt {x}}{3 (25 a x-6)}}+\frac {\sqrt {6}}{5 \sqrt {a} \sqrt {x}}\right )^2\right )}{3 \sqrt [6]{1-\left (\frac {1}{\frac {\sqrt {\frac {2}{3}} \sqrt {a} \sqrt [9]{x} \left (25 a x (25 a x-6)^{7/18}-6 (25 a x-6)^{7/18}\right ) y(x)}{25\ 5^{7/9}}+\frac {25 \sqrt {\frac {2}{3}} \sqrt {a} \sqrt {x}}{3 (25 a x-6)}}+\frac {\sqrt {6}}{5 \sqrt {a} \sqrt {x}}\right )^2}}-\frac {5 \sqrt {a} \sqrt {x}}{\sqrt {6}}\right ),y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) + (a*x - 6/25)**(34/9)*y(x)**3/x**(16/9) - (2*a*x/3 - 4/675)/(x**(11/18)*(a*x - 6/25)**(61/18)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out