11.5.7 problem 7

Internal problem ID [1512]
Book : Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima, Meade
Section : Chapter 6.5, The Laplace Transform. Impulse functions. page 273
Problem number : 7
Date solved : Saturday, March 29, 2025 at 10:57:08 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+2 y&=\cos \left (t \right )+\delta \left (t -\frac {\pi }{2}\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: 0.425 (sec). Leaf size: 46

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

\[ y = \frac {\cos \left (t \right )}{5}+\frac {2 \sin \left (t \right )}{5}+\frac {\left (-\cos \left (t \right )-3 \sin \left (t \right )\right ) {\mathrm e}^{-t}}{5}-\cos \left (t \right ) \operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) {\mathrm e}^{-t +\frac {\pi }{2}} \]

✓ Mathematica. Time used: 0.16 (sec). Leaf size: 52

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

\[ y(t)\to \frac {1}{5} e^{-t} \left (-5 e^{\pi /2} \theta (2 t-\pi ) \cos (t)+\left (2 e^t-3\right ) \sin (t)+\left (e^t-1\right ) \cos (t)\right ) \]

✓ Sympy. Time used: 9.519 (sec). Leaf size: 102

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Dirac(t - pi/2) + 2*y(t) - cos(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)
 

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