61.4.31 problem Problem 5(b)

Internal problem ID [15356]
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(b)
Date solved : Thursday, October 02, 2025 at 10:12:02 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+2 y&=3 \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: 0.105 (sec). Leaf size: 20

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

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

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 95

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

\begin{align*} y(t)&\to -e^{-t} \left (\sin (t) \int _1^03 e \cos (1) \delta (K[1]-1)dK[1]-\sin (t) \int _1^t3 e \cos (1) \delta (K[1]-1)dK[1]+\cos (t) \int _1^0-3 e \delta (K[2]-1) \sin (1)dK[2]-\cos (t) \int _1^t-3 e \delta (K[2]-1) \sin (1)dK[2]\right ) \end{align*}

✓ Sympy. Time used: 1.468 (sec). Leaf size: 73

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-3*Dirac(t - 1) + 2*y(t) + 2*Derivative(y(t), t) + 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)
 

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