22.3.15 problem 6.50

Internal problem ID [4528]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 6. The Laplace Transform and Its Applications. Problems at page 291
Problem number : 6.50
Date solved : Tuesday, September 30, 2025 at 07:33:55 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=4 \operatorname {Heaviside}\left (t -\pi \right )+2 \delta \left (t -\pi \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.379 (sec). Leaf size: 33

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

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

✓ Mathematica. Time used: 0.108 (sec). Leaf size: 40

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

\begin{align*} y(t)&\to \theta (t-\pi ) \sin (2 t)-2 \theta (\pi -t) \sin ^2(t)+\sin (2 t)-2 \cos (2 t)+1 \end{align*}

✓ Sympy. Time used: 1.254 (sec). Leaf size: 110

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

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