55.26.9 problem 9

Internal problem ID [13756]
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 : 9
Date solved : Thursday, October 02, 2025 at 07:58:08 AM
CAS classification : [[_homogeneous, `class G`], _Abel]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 36

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

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

✓ Mathematica. Time used: 0.509 (sec). Leaf size: 403

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

\[ \text {Solve}\left [\frac {1}{3} a b^2 \text {RootSum}\left [\text {$\#$1}^9 a b^2+3 \text {$\#$1}^6 a b^2+3 \text {$\#$1}^3 a b^2+\text {$\#$1}^3 n^3+3 \text {$\#$1}^3 n^2+3 \text {$\#$1}^3 n+\text {$\#$1}^3+a b^2\&,\frac {\text {$\#$1}^6 \log \left (y(x) \sqrt [3]{\frac {a x^{3 n+3}}{b}}-\text {$\#$1}\right )+\text {$\#$1}^4 \sqrt [3]{-\frac {(n+1)^3}{a b^2}} \log \left (y(x) \sqrt [3]{\frac {a x^{3 n+3}}{b}}-\text {$\#$1}\right )+2 \text {$\#$1}^3 \log \left (y(x) \sqrt [3]{\frac {a x^{3 n+3}}{b}}-\text {$\#$1}\right )+\text {$\#$1}^2 \left (-\frac {(n+1)^3}{a b^2}\right )^{2/3} \log \left (y(x) \sqrt [3]{\frac {a x^{3 n+3}}{b}}-\text {$\#$1}\right )+\text {$\#$1} \sqrt [3]{-\frac {(n+1)^3}{a b^2}} \log \left (y(x) \sqrt [3]{\frac {a x^{3 n+3}}{b}}-\text {$\#$1}\right )+\log \left (y(x) \sqrt [3]{\frac {a x^{3 n+3}}{b}}-\text {$\#$1}\right )}{3 \text {$\#$1}^8 a b^2+6 \text {$\#$1}^5 a b^2+3 \text {$\#$1}^2 a b^2+\text {$\#$1}^2 n^3+3 \text {$\#$1}^2 n^2+3 \text {$\#$1}^2 n+\text {$\#$1}^2}\&\right ]=\int _1^xb K[1]^{-n-2} \sqrt [3]{\frac {a K[1]^{3 n+3}}{b}}dK[1]+c_1,y(x)\right ] \]

✗ Sympy

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

NotImplementedError : Cannot solve the partial differential equation due to inability of constantsim