67.4.33 problem Problem 5(d)

Internal problem ID [13998]
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(d)
Date solved : Wednesday, March 05, 2025 at 10:25:12 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+2 y&=1-\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: 9.613 (sec). Leaf size: 41

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

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

✓ Mathematica. Time used: 0.07 (sec). Leaf size: 106

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

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

✓ Sympy. Time used: 1.632 (sec). Leaf size: 71

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

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