89.1.14 problem 14

Internal problem ID [24249]
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 21
Problem number : 14
Date solved : Thursday, October 02, 2025 at 10:01:54 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 71

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

\begin{align*} y &= -\frac {-1+\sqrt {2 x \,{\mathrm e}^{x}-2 \,{\mathrm e}^{x}+2 c_1 +1}}{2 \left (x \,{\mathrm e}^{x}-{\mathrm e}^{x}+c_1 \right )} \\ y &= \frac {1+\sqrt {2 x \,{\mathrm e}^{x}-2 \,{\mathrm e}^{x}+2 c_1 +1}}{2 x \,{\mathrm e}^{x}+2 c_1 -2 \,{\mathrm e}^{x}} \\ \end{align*}

✓ Mathematica. Time used: 9.668 (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: 0.759 (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)
 

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