85.71.2 problem 5 (a)

Internal problem ID [22934]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 6. Solution of linear differential equations by Laplace transform. C Exercises at page 284
Problem number : 5 (a)
Date solved : Thursday, October 02, 2025 at 09:16:44 PM
CAS classification : [[_linear, `class A`]]

\begin{align*} y^{\prime }+2 y&=5 \delta \left (t -1\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.093 (sec). Leaf size: 23

ode:=diff(y(t),t)+2*y(t) = 5*Dirac(t-1); 
ic:=[y(0) = 2]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = 5 \operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{-2 t +2}+2 \,{\mathrm e}^{-2 t} \]

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 22

ode=D[y[t],t]+2*y[t]==5*DiracDelta[t-1]; 
ic={y[0]==2}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{-2 t} \left (5 e^2 \theta (t-1)+2\right ) \end{align*}

✓ Sympy. Time used: 0.485 (sec). Leaf size: 49

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-5*Dirac(t - 1) + 2*y(t) + Derivative(y(t), t),0) 
ics = {y(0): 2} 
dsolve(ode,func=y(t),ics=ics)
 

\[ - 5 \int \operatorname {Dirac}{\left (t - 1 \right )} e^{2 t}\, dt + 2 \int y{\left (t \right )} e^{2 t}\, dt = - 5 \int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} e^{2 t}\, dt + 2 \int \limits ^{0} y{\left (t \right )} e^{2 t}\, dt \]