54.1.244 problem 249

Internal problem ID [11558]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 249
Date solved : Tuesday, September 30, 2025 at 09:00:56 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`], [_Abel, `2nd type`, `class C`]]

\begin{align*} \left (a x y+b \,x^{n}\right ) y^{\prime }+\alpha y^{3}+\beta y^{2}&=0 \end{align*}

✓ Maple. Time used: 0.129 (sec). Leaf size: 202

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

\[ y = \frac {\beta }{\operatorname {RootOf}\left (-\textit {\_Z}^{\frac {a \left (n -1\right )}{\beta }} x^{-n +1} a^{2} \beta n +c_1 \,a^{2} b \,n^{2}-\textit {\_Z}^{\frac {a n -a +\beta }{\beta }} \beta b a n +\textit {\_Z}^{\frac {a \left (n -1\right )}{\beta }} x^{-n +1} a^{2} \beta -\textit {\_Z}^{\frac {a \left (n -1\right )}{\beta }} x^{-n +1} a \,\beta ^{2}+\textit {\_Z}^{\frac {a \left (n -1\right )}{\beta }} a \alpha b n -2 c_1 \,a^{2} b n +c_1 a b \beta n +\textit {\_Z}^{\frac {a n -a +\beta }{\beta }} \beta b a -\textit {\_Z}^{\frac {a \left (n -1\right )}{\beta }} a \alpha b +\textit {\_Z}^{\frac {a \left (n -1\right )}{\beta }} \alpha b \beta +c_1 \,a^{2} b -c_1 a b \beta \right ) \beta -\alpha } \]

✓ Mathematica. Time used: 1.478 (sec). Leaf size: 299

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

\[ \text {Solve}\left [\int _1^{-\frac {b x^{n-1} \left (b (a (n-1)-\beta ) x^n+\left (-3 b \alpha x^n+a^2 (n-1) x+2 a \beta x\right ) y(x)\right )}{a \sqrt [3]{-\frac {b^3 x^{3 n-3} \left (2 a^3 (n-1)^3+3 a^2 \beta (n-1)^2-3 a \beta ^2 (n-1)-2 \beta ^3\right )}{a^3}} \left (b x^n+a y(x) x\right )}}\frac {1}{K[1]^3+\frac {3 \sqrt [3]{-1} \left (a^2 (n-1)^2+a \beta (n-1)+\beta ^2\right ) K[1]}{(a (n-1)-\beta )^{2/3} (2 a (n-1)+\beta )^{2/3} (a (n-1)+2 \beta )^{2/3}}+1}dK[1]+\frac {a x^{2-2 n} \left (-\frac {b^3 x^{3 n-3} \left (2 a^3 (n-1)^3+3 a^2 \beta (n-1)^2-3 a \beta ^2 (n-1)-2 \beta ^3\right )}{a^3}\right )^{2/3} \log \left (\alpha b-a \beta x^{1-n}\right )}{9 b^2 \beta (n-1)}=c_1,y(x)\right ] \]

✗ Sympy

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

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