36.3.12 problem 13

Internal problem ID [6333]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 2, First order differential equations. Section 2.4, Exact equations. Exercises. page 64
Problem number : 13
Date solved : Sunday, March 30, 2025 at 10:51:56 AM
CAS classification : [_linear]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 19

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

\[ y = \frac {\left (t -1\right ) {\mathrm e}^{t}+c_1}{1+{\mathrm e}^{t}} \]

✓ Mathematica. Time used: 0.081 (sec). Leaf size: 23

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

\[ y(t)\to \frac {e^t (t-1)+c_1}{e^t+1} \]

✓ Sympy. Time used: 0.297 (sec). Leaf size: 17

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

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