55.26.16 problem 16

Internal problem ID [13763]
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 : 16
Date solved : Thursday, October 02, 2025 at 07:59:57 AM
CAS classification : [_rational, _Abel]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 54

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

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

✓ Mathematica. Time used: 0.209 (sec). Leaf size: 321

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

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

✗ Sympy

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

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