76.2.3 problem 3

Internal problem ID [17260]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 2. First order differential equations. Section 2.2 (Linear equations: Method of integrating factors). Problems at page 54
Problem number : 3
Date solved : Thursday, March 13, 2025 at 09:22:15 AM
CAS classification : [[_linear, `class A`]]

\begin{align*} y^{\prime }+y&=t \,{\mathrm e}^{-t}+1 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 19

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

\[ y = 1+\frac {\left (t^{2}+2 c_{1} \right ) {\mathrm e}^{-t}}{2} \]

✓ Mathematica. Time used: 0.06 (sec). Leaf size: 27

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

\[ y(t)\to \frac {1}{2} e^{-t} \left (t^2+2 e^t+2 c_1\right ) \]

✓ Sympy. Time used: 0.167 (sec). Leaf size: 14

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

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