76.2.14 problem 14

Internal problem ID [17279]
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 : 14
Date solved : Monday, March 31, 2025 at 03:48:32 PM
CAS classification : [[_linear, `class A`]]

\begin{align*} y^{\prime }+2 y&=t \,{\mathrm e}^{-2 t} \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=0 \end{align*}

✓ Maple. Time used: 0.027 (sec). Leaf size: 15

ode:=diff(y(t),t)+2*y(t) = exp(-2*t)*t; 
ic:=y(1) = 0; 
dsolve([ode,ic],y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.061 (sec). Leaf size: 19

ode=D[y[t],t]+2*y[t]==t*Exp[-2*t]; 
ic={y[1]==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.180 (sec). Leaf size: 15

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t*exp(-2*t) + 2*y(t) + Derivative(y(t), t),0) 
ics = {y(1): 0} 
dsolve(ode,func=y(t),ics=ics)
 

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