72.20.2 problem 3

Internal problem ID [14889]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 6. Laplace transform. Section 6.4. page 608
Problem number : 3
Date solved : Thursday, March 13, 2025 at 05:18:33 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+5 y&=\delta \left (t -3\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Maple. Time used: 11.059 (sec). Leaf size: 37

ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)+5*y(t) = Dirac(t-3); 
ic:=y(0) = 1, D(y)(0) = 1; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = {\mathrm e}^{-t} \left (\sin \left (2 t \right )+\cos \left (2 t \right )\right )+\frac {{\mathrm e}^{-t +3} \operatorname {Heaviside}\left (t -3\right ) \sin \left (2 t -6\right )}{2} \]

✓ Mathematica. Time used: 0.072 (sec). Leaf size: 131

ode=D[y[t],{t,2}]+2*D[y[t],t]+5*y[t]==DiracDelta[t-3]; 
ic={y[0]==1,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to -e^{-t} \left (\cos (2 t) \int _1^0-e^3 \cos (3) \delta (K[2]-3) \sin (3)dK[2]-\cos (2 t) \int _1^t-e^3 \cos (3) \delta (K[2]-3) \sin (3)dK[2]+\sin (2 t) \int _1^0\frac {1}{2} e^3 \cos (6) \delta (K[1]-3)dK[1]-\sin (2 t) \int _1^t\frac {1}{2} e^3 \cos (6) \delta (K[1]-3)dK[1]-\sin (2 t)-\cos (2 t)\right ) \]

✓ Sympy. Time used: 4.573 (sec). Leaf size: 87

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Dirac(t - 3) + 5*y(t) + 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\left (- \frac {\int \operatorname {Dirac}{\left (t - 3 \right )} e^{t} \sin {\left (2 t \right )}\, dt}{2} + \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - 3 \right )} e^{t} \sin {\left (2 t \right )}\, dt}{2} + 1\right ) \cos {\left (2 t \right )} + \left (\frac {\int \operatorname {Dirac}{\left (t - 3 \right )} e^{t} \cos {\left (2 t \right )}\, dt}{2} - \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - 3 \right )} e^{t} \cos {\left (2 t \right )}\, dt}{2} + 1\right ) \sin {\left (2 t \right )}\right ) e^{- t} \]