83.3.9 problem 9

Internal problem ID [18986]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter II. Equations of first order and first degree. Exercise II (B) at page 9
Problem number : 9
Date solved : Thursday, March 13, 2025 at 01:16:53 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 22

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

\[ y \left (x \right ) = -\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-1}}{c_{1} \left ({\mathrm e}^{x}+1\right )}\right )-1 \]

✓ Mathematica. Time used: 3.771 (sec). Leaf size: 32

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

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

✓ Sympy. Time used: 0.392 (sec). Leaf size: 19

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

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