54.1.185 problem 188

Internal problem ID [11499]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 188
Date solved : Tuesday, September 30, 2025 at 08:47:47 PM
CAS classification : [[_homogeneous, `class G`], _Abel]

\begin{align*} x^{2 n +1} y^{\prime }-a y^{3}-b \,x^{3 n}&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 32

ode:=x^(2*n+1)*diff(y(x),x)-a*y(x)^3-b*x^(3*n) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.195 (sec). Leaf size: 331

ode=x^(2*n+1)*D[y[x],x] - a*y[x]^3 - b*x^(3*n)==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

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

NotImplementedError : The given ODE -a*x**(-2*n - 1)*y(x)**3 - b*x**(n - 1) + Derivative(y(x), x) ca