20.23.3 problem Problem 3

Internal problem ID [3975]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 10, The Laplace Transform and Some Elementary Applications. Exercises for 10.8. page 710
Problem number : Problem 3
Date solved : Sunday, March 30, 2025 at 02:13:28 AM
CAS classification : [[_linear, `class A`]]

\begin{align*} y^{\prime }+4 y&=3 \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: 0.107 (sec). Leaf size: 23

ode:=diff(y(t),t)+4*y(t) = 3*Dirac(t-1); 
ic:=y(0) = 2; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = 3 \,{\mathrm e}^{4} \operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{-4 t}+2 \,{\mathrm e}^{-4 t} \]

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 22

ode=D[y[t],t]+4*y[t]==3*DiracDelta[t-1]; 
ic={y[0]==2}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to e^{-4 t} \left (3 e^4 \theta (t-1)+2\right ) \]

✓ Sympy. Time used: 0.771 (sec). Leaf size: 49

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-3*Dirac(t - 1) + 4*y(t) + Derivative(y(t), t),0) 
ics = {y(0): 2} 
dsolve(ode,func=y(t),ics=ics)
 

\[ - 3 \int \operatorname {Dirac}{\left (t - 1 \right )} e^{4 t}\, dt + 4 \int y{\left (t \right )} e^{4 t}\, dt = - 3 \int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} e^{4 t}\, dt + 4 \int \limits ^{0} y{\left (t \right )} e^{4 t}\, dt \]