55.26.15 problem 15

Internal problem ID [13762]
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 : 15
Date solved : Thursday, October 02, 2025 at 07:59:54 AM
CAS classification : [_rational, _Abel]

\begin{align*} x y^{\prime }&=3 x^{1+2 n} y^{3}+\left (b x -n \right ) y+c \,x^{1-n} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 51

ode:=x*diff(y(x),x) = 3*x^(2*n+1)*y(x)^3+(b*x-n)*y(x)+c*x^(-n+1); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {\operatorname {RootOf}\left (-b^{3} \int _{}^{\textit {\_Z}}\frac {1}{3 \textit {\_a}^{3} c^{2}+\textit {\_a} \,b^{3}-b^{3}}d \textit {\_a} +b x +c_1 \right ) c \,x^{-n}}{b} \]

✓ Mathematica. Time used: 0.325 (sec). Leaf size: 365

ode=x*D[y[x],x]==3*x^(2*n+1)*y[x]^3+(b*x-n)*y[x]+c*x^(1-n); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\frac {1}{3} c^2 \text {RootSum}\left [3 \text {$\#$1}^9 c^2+9 \text {$\#$1}^6 c^2+\text {$\#$1}^3 b^3+9 \text {$\#$1}^3 c^2+3 c^2\&,\frac {3 \text {$\#$1}^6 \log \left (\sqrt [3]{3} y(x) \sqrt [3]{\frac {x^{3 n}}{c}}-\text {$\#$1}\right )+3^{2/3} \text {$\#$1}^4 \sqrt [3]{-\frac {b^3}{c^2}} \log \left (\sqrt [3]{3} y(x) \sqrt [3]{\frac {x^{3 n}}{c}}-\text {$\#$1}\right )+6 \text {$\#$1}^3 \log \left (\sqrt [3]{3} y(x) \sqrt [3]{\frac {x^{3 n}}{c}}-\text {$\#$1}\right )+\sqrt [3]{3} \text {$\#$1}^2 \left (-\frac {b^3}{c^2}\right )^{2/3} \log \left (\sqrt [3]{3} y(x) \sqrt [3]{\frac {x^{3 n}}{c}}-\text {$\#$1}\right )+3^{2/3} \text {$\#$1} \sqrt [3]{-\frac {b^3}{c^2}} \log \left (\sqrt [3]{3} y(x) \sqrt [3]{\frac {x^{3 n}}{c}}-\text {$\#$1}\right )+3 \log \left (\sqrt [3]{3} y(x) \sqrt [3]{\frac {x^{3 n}}{c}}-\text {$\#$1}\right )}{9 \text {$\#$1}^8 c^2+18 \text {$\#$1}^5 c^2+\text {$\#$1}^2 b^3+9 \text {$\#$1}^2 c^2}\&\right ]=\sqrt [3]{3} c x^{1-n} \sqrt [3]{\frac {x^{3 n}}{c}}+c_1,y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
n = symbols("n") 
c = symbols("c") 
y = Function("y") 
ode = Eq(-c*x**(1 - n) + x*Derivative(y(x), x) - 3*x**(2*n + 1)*y(x)**3 - (b*x - n)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (b*x*y(x) + c*x**(1 - n) - n*y(x) + 3*x**(2*n + 1)*y(x)**3)/x cannot be solved by the factorable group method