73.22.5 problem 31.6 (e)

Internal problem ID [15553]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 31. Delta Functions. Additional Exercises. page 572
Problem number : 31.6 (e)
Date solved : Thursday, March 13, 2025 at 06:11:22 AM
CAS classification : [[_linear, `class A`]]

\begin{align*} y^{\prime }+2 y&=4 \delta \left (t -1\right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 9.559 (sec). Leaf size: 16

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

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

✓ Mathematica. Time used: 0.03 (sec). Leaf size: 28

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

\[ y(t)\to e^{-2 t} \int _0^t4 e^2 \delta (K[1]-1)dK[1] \]

✓ Sympy. Time used: 0.740 (sec). Leaf size: 49

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

\[ - 4 \int \operatorname {Dirac}{\left (t - 1 \right )} e^{2 t}\, dt + 2 \int y{\left (t \right )} e^{2 t}\, dt = - 4 \int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} e^{2 t}\, dt + 2 \int \limits ^{0} y{\left (t \right )} e^{2 t}\, dt \]