20.23.7 problem Problem 7

Internal problem ID [3979]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 10, The Laplace Transform and Some Elementary Applications. Exercises for 10.8. page 710
Problem number : Problem 7
Date solved : Sunday, March 30, 2025 at 02:13:35 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+5 y&=\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 )&=2 \end{align*}

✓ Maple. Time used: 0.311 (sec). Leaf size: 27

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

\[ y = {\mathrm e}^{-t} \left (-\frac {\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) {\mathrm e}^{\frac {\pi }{2}}}{2}+1\right ) \sin \left (2 t \right ) \]

✓ Mathematica. Time used: 0.116 (sec). Leaf size: 34

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

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

✓ Sympy. Time used: 5.434 (sec). Leaf size: 92

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

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