60.1.186 problem 189

Internal problem ID [10200]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 189
Date solved : Sunday, March 30, 2025 at 03:28:49 PM
CAS classification : [[_homogeneous, `class G`]]

\begin{align*} x^{m \left (n -1\right )+n} y^{\prime }-a y^{n}-b \,x^{n \left (m +1\right )}&=0 \end{align*}

✓ Maple. Time used: 0.107 (sec). Leaf size: 62

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

\[ -x^{n \left (m +1\right )} \int _{\textit {\_b}}^{y}\frac {1}{b \,x^{\left (n +1\right ) \left (m +1\right )}-\textit {\_a} \left (m +1\right ) x^{n \left (m +1\right )}+a \,x^{m +1} \textit {\_a}^{n}}d \textit {\_a} +\ln \left (x \right )-c_1 = 0 \]

✓ Mathematica. Time used: 0.886 (sec). Leaf size: 91

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

\[ \text {Solve}\left [\int _1^{\left (\frac {a x^{-((m+1) n)}}{b}\right )^{\frac {1}{n}} y(x)}\frac {1}{K[1]^n-\left (\frac {b^{1-n} (m+1)^n}{a}\right )^{\frac {1}{n}} K[1]+1}dK[1]=b x^{m+1} \log (x) \left (\frac {a x^{-((m+1) n)}}{b}\right )^{\frac {1}{n}}+c_1,y(x)\right ] \]

✗ Sympy

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

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