78.4.6 problem 4.a

Internal problem ID [21034]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 5, Laplace transforms. Problems section 5.7
Problem number : 4.a
Date solved : Thursday, October 02, 2025 at 07:01:37 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=\delta \left (t -1\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=3 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.276 (sec). Leaf size: 23

ode:=diff(diff(y(t),t),t)+4*y(t) = Dirac(t-1); 
ic:=[y(0) = 3, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \frac {\operatorname {Heaviside}\left (t -1\right ) \sin \left (-2+2 t \right )}{2}+3 \cos \left (2 t \right ) \]

✓ Mathematica. Time used: 0.068 (sec). Leaf size: 26

ode=D[y[t],{t,2}]+4*y[t]==DiracDelta[t-1]; 
ic={y[0]==3,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to 3 \cos (2 t)-\frac {1}{2} \theta (t-1) \sin (2-2 t) \end{align*}

✓ Sympy. Time used: 0.712 (sec). Leaf size: 68

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Dirac(t - 1) + 4*y(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 3, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\frac {\int \operatorname {Dirac}{\left (t - 1 \right )} \cos {\left (2 t \right )}\, dt}{2} - \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} \cos {\left (2 t \right )}\, dt}{2}\right ) \sin {\left (2 t \right )} + \left (- \frac {\int \operatorname {Dirac}{\left (t - 1 \right )} \sin {\left (2 t \right )}\, dt}{2} + \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} \sin {\left (2 t \right )}\, dt}{2} + 3\right ) \cos {\left (2 t \right )} \]