70.21.5 problem 5

Internal problem ID [19055]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.7 (Impulse Functions). Problems at page 350
Problem number : 5
Date solved : Thursday, October 02, 2025 at 03:37:32 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+3 y&=\sin \left (t \right )+\delta \left (t -3 \pi \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: 0.390 (sec). Leaf size: 54

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

\[ y = \frac {{\mathrm e}^{3 \pi -t} \sqrt {2}\, \operatorname {Heaviside}\left (t -3 \pi \right ) \sin \left (\sqrt {2}\, \left (t -3 \pi \right )\right )}{2}+\frac {{\mathrm e}^{-t} \cos \left (\sqrt {2}\, t \right )}{4}-\frac {\cos \left (t \right )}{4}+\frac {\sin \left (t \right )}{4} \]

✓ Mathematica. Time used: 0.534 (sec). Leaf size: 201

ode=D[y[t],{t,2}]+2*D[y[t],t]+3*y[t]==Sin[t]+DiracDelta[t-3*Pi]; 
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 \left (\sqrt {2} t\right ) \int _1^0\frac {e^{K[1]} \cos \left (\sqrt {2} K[1]\right ) (\delta (K[1]-3 \pi )+\sin (K[1]))}{\sqrt {2}}dK[1]-\sin \left (\sqrt {2} t\right ) \int _1^t\frac {e^{K[1]} \cos \left (\sqrt {2} K[1]\right ) (\delta (K[1]-3 \pi )+\sin (K[1]))}{\sqrt {2}}dK[1]+\cos \left (\sqrt {2} t\right ) \int _1^0-\frac {e^{K[2]} (\delta (K[2]-3 \pi )+\sin (K[2])) \sin \left (\sqrt {2} K[2]\right )}{\sqrt {2}}dK[2]-\cos \left (\sqrt {2} t\right ) \int _1^t-\frac {e^{K[2]} (\delta (K[2]-3 \pi )+\sin (K[2])) \sin \left (\sqrt {2} K[2]\right )}{\sqrt {2}}dK[2]\right ) \end{align*}

✓ Sympy. Time used: 94.970 (sec). Leaf size: 185

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Dirac(t - 3*pi) + 3*y(t) - sin(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)
 

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