67.4.34 problem Problem 5(e)

Internal problem ID [14007]
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 5(e)
Date solved : Monday, March 31, 2025 at 08:21:41 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.176 (sec). Leaf size: 17

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

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

✓ Mathematica. Time used: 0.03 (sec). Leaf size: 86

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

\[ y(t)\to -e^{-t/2} \left (-\int _1^t-\frac {1}{4} \delta (K[1]-1)dK[1]+t \int _1^0\frac {1}{4} \delta (K[2]-1)dK[2]-t \int _1^t\frac {1}{4} \delta (K[2]-1)dK[2]+\int _1^0-\frac {1}{4} \delta (K[1]-1)dK[1]\right ) \]

✓ Sympy. Time used: 0.651 (sec). Leaf size: 46

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

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