76.21.6 problem 6

Internal problem ID [17699]
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 : 6
Date solved : Monday, March 31, 2025 at 04:25:14 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&={\frac {1}{2}}\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 0.203 (sec). Leaf size: 23

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

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

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 25

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

\[ y(t)\to \frac {1}{2} (\theta (t-4 \pi ) \sin (2 t)+\cos (2 t)) \]

✓ Sympy. Time used: 1.217 (sec). Leaf size: 76

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

\[ y{\left (t \right )} = \left (\frac {\int \operatorname {Dirac}{\left (t - 4 \pi \right )} \cos {\left (2 t \right )}\, dt}{2} - \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - 4 \pi \right )} \cos {\left (2 t \right )}\, dt}{2}\right ) \sin {\left (2 t \right )} + \left (- \frac {\int \operatorname {Dirac}{\left (t - 4 \pi \right )} \sin {\left (2 t \right )}\, dt}{2} + \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - 4 \pi \right )} \sin {\left (2 t \right )}\, dt}{2} + \frac {1}{2}\right ) \cos {\left (2 t \right )} \]