28.1.7 problem 7

Internal problem ID [4313]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page 78
Problem number : 7
Date solved : Tuesday, March 04, 2025 at 06:20:30 PM
CAS classification : [_separable]

\begin{align*} x y^{3}+\left (y+1\right ) {\mathrm e}^{-x} y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 73

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

\begin{align*} y \left (x \right ) &= \frac {1-\sqrt {\left (2 x -2\right ) {\mathrm e}^{x}+2 c_{1} +1}}{\left (2 x -2\right ) {\mathrm e}^{x}+2 c_{1}} \\ y \left (x \right ) &= \frac {\sqrt {\left (2 x -2\right ) {\mathrm e}^{x}+2 c_{1} +1}+1}{\left (2 x -2\right ) {\mathrm e}^{x}+2 c_{1}} \\ \end{align*}

✓ Mathematica. Time used: 9.928 (sec). Leaf size: 88

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

\begin{align*} y(x)\to \frac {1-\sqrt {2 e^x (x-1)+1-2 c_1}}{2 e^x (x-1)-2 c_1} \\ y(x)\to \frac {1+\sqrt {2 e^x (x-1)+1-2 c_1}}{2 e^x (x-1)-2 c_1} \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 1.120 (sec). Leaf size: 78

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

\[ \left [ y{\left (x \right )} = \frac {\sqrt {- 2 C_{1} + 2 x e^{x} - 2 e^{x} + 1} - 1}{2 \left (C_{1} - x e^{x} + e^{x}\right )}, \ y{\left (x \right )} = \frac {\sqrt {- 2 C_{1} + 2 x e^{x} - 2 e^{x} + 1} + 1}{2 \left (- C_{1} + x e^{x} - e^{x}\right )}\right ] \]