61.4.9 problem Problem 2(i)

Internal problem ID [15334]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 5.6 Laplace transform. Nonhomogeneous equations. Problems page 368
Problem number : Problem 2(i)
Date solved : Thursday, October 02, 2025 at 10:11:43 AM
CAS classification : [[_linear, `class A`]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.087 (sec). Leaf size: 13

ode:=2*diff(y(t),t)+y(t) = exp(-1/2*t); 
ic:=[y(0) = -1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.033 (sec). Leaf size: 19

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

\begin{align*} y(t)&\to \frac {1}{2} e^{-t/2} (t-2) \end{align*}

✓ Sympy. Time used: 0.116 (sec). Leaf size: 12

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

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