89.4.27 problem 28

Internal problem ID [24349]
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 : 28
Date solved : Thursday, October 02, 2025 at 10:21:00 PM
CAS classification : [[_homogeneous, `class G`], _rational]

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

✓ Maple. Time used: 0.014 (sec). Leaf size: 50

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

\[ y = x^{-\frac {a}{b}} {\mathrm e}^{\frac {-\operatorname {LambertW}\left (\frac {x^{-\frac {\left (a -b \right ) n}{b}} {\mathrm e}^{\frac {c_1}{b}}}{b}\right ) b +c_1}{b n}} \]

✓ Mathematica. Time used: 60.444 (sec). Leaf size: 48

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

\begin{align*} y(x)&\to \left (b x^{-n} W\left (\frac {x^n \left (e^{\frac {c_1}{b}} \left (x^n\right )^{-\frac {a}{b n}}\right ){}^n}{b}\right )\right ){}^{\frac {1}{n}} \end{align*}

✗ Sympy

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

Timed Out