89.4.28 problem 29

Internal problem ID [24350]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Exercises at page 39
Problem number : 29
Date solved : Thursday, October 02, 2025 at 10:21:07 PM
CAS classification : [_rational]

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

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.56 (sec). Leaf size: 65

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

\[ \text {Solve}\left [\frac {(x y(x))^{-n} \left ((n-1) x^{n+2} y(x)^n-2 a x y(x)\right )}{2 (n-1)}+\frac {1}{2} x^n y(x)^{n+2} (x y(x))^{-n}=c_1,y(x)\right ] \]

✗ Sympy

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

Timed Out