76.21.7 problem 7

Internal problem ID [17692]
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 : 7
Date solved : Thursday, March 13, 2025 at 10:47:17 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\delta \left (t -2 \pi \right ) \cos \left (t \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 10.319 (sec). Leaf size: 15

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

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

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 16

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

\[ y(t)\to (\theta (t-2 \pi )+1) \sin (t) \]

✓ Sympy. Time used: 1.220 (sec). Leaf size: 68

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

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