76.21.8 problem 8

Internal problem ID [17693]
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 : 8
Date solved : Thursday, March 13, 2025 at 10:47:18 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=2 \delta \left (t -\frac {\pi }{4}\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: 13.216 (sec). Leaf size: 16

ode:=diff(diff(y(t),t),t)+4*y(t) = 2*Dirac(t-1/4*Pi); 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = -\operatorname {Heaviside}\left (t -\frac {\pi }{4}\right ) \cos \left (2 t \right ) \]

✓ Mathematica. Time used: 0.029 (sec). Leaf size: 19

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

\[ y(t)\to \theta (4 t-\pi ) (-\cos (2 t)) \]

✓ Sympy. Time used: 1.164 (sec). Leaf size: 66

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-2*Dirac(t - pi/4) + 4*y(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)
 

\[ y{\left (t \right )} = \left (- \int \operatorname {Dirac}{\left (t - \frac {\pi }{4} \right )} \sin {\left (2 t \right )}\, dt + \int \limits ^{0} \operatorname {Dirac}{\left (t - \frac {\pi }{4} \right )} \sin {\left (2 t \right )}\, dt\right ) \cos {\left (2 t \right )} + \left (\int \operatorname {Dirac}{\left (t - \frac {\pi }{4} \right )} \cos {\left (2 t \right )}\, dt - \int \limits ^{0} \operatorname {Dirac}{\left (t - \frac {\pi }{4} \right )} \cos {\left (2 t \right )}\, dt\right ) \sin {\left (2 t \right )} \]