46.8.11 problem 11

Internal problem ID [9661]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 7 THE LAPLACE TRANSFORM. EXERCISES 7.5. Page 315
Problem number : 11
Date solved : Tuesday, September 30, 2025 at 06:22:00 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+13 y&=\delta \left (t -\pi \right )+\delta \left (t -3 \pi \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.341 (sec). Leaf size: 53

ode:=diff(diff(y(t),t),t)+4*diff(y(t),t)+13*y(t) = Dirac(t-Pi)+Dirac(t-3*Pi); 
ic:=[y(0) = 1, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.082 (sec). Leaf size: 113

ode=D[y[t],{t,2}]+4*D[y[t],t]+13*y[t]==DiracDelta[t-Pi]+DiracDelta[t-3*Pi]; 
ic={y[0]==1,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 5.936 (sec). Leaf size: 151

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Dirac(t - 3*pi) - Dirac(t - pi) + 13*y(t) + 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

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