11.5.11 problem 12
Internal
problem
ID
[1516]
Book
:
Elementary
differential
equations
and
boundary
value
problems,
11th
ed.,
Boyce,
DiPrima,
Meade
Section
:
Chapter
6.5,
The
Laplace
Transform.
Impulse
functions.
page
273
Problem
number
:
12
Date
solved
:
Saturday, March 29, 2025 at 10:57:17 PM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime }+y&=\frac {\operatorname {Heaviside}\left (t -4+k \right )-\operatorname {Heaviside}\left (t -4-k \right )}{2 k} \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: 0.330 (sec). Leaf size: 83
ode:=diff(diff(y(t),t),t)+y(t) = 1/2/k*(Heaviside(t-4+k)-Heaviside(t-4-k));
ic:=y(0) = 0, D(y)(0) = 0;
dsolve([ode,ic],y(t),method='laplace');
\[
y = \frac {\left (\operatorname {Heaviside}\left (4+k \right )+\operatorname {Heaviside}\left (t -4-k \right )-1\right ) \cos \left (-t +4+k \right )-\operatorname {Heaviside}\left (t -4-k \right )+\left (-\cos \left (t -4+k \right )+1\right ) \operatorname {Heaviside}\left (t -4+k \right )-\operatorname {Heaviside}\left (-4+k \right ) \cos \left (t \right )-\cos \left (t \right ) \operatorname {Heaviside}\left (4+k \right )+\operatorname {Heaviside}\left (-4+k \right ) \cos \left (t -4+k \right )+\cos \left (t \right )}{2 k}
\]
✓ Mathematica. Time used: 1.081 (sec). Leaf size: 181
ode=D[y[t],{t,2}]+y[t]==1/(2*k)*(UnitStep[t-(4-k)] - UnitStep[t-(4+k)] );
ic={y[0]==0,Derivative[1][y][0] ==0};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\begin{align*}
y(t)\to \fbox {$\frac {(\cos (k-t+4)-1) \theta (-k+t-4)-(\cos (-k-t+4)-1) \theta (k+t-4)}{2 k}\text { if }-4<k<4$} \\
y(t)\to \fbox {$\frac {\cos (-k-t+4)-\cos (t)+(\cos (k-t+4)-1) \theta (-k+t-4)-(\cos (-k-t+4)-1) \theta (k+t-4)}{2 k}\text { if }k>4$} \\
y(t)\to \fbox {$\frac {-\cos (k-t+4)+\cos (t)+(\cos (k-t+4)-1) \theta (-k+t-4)-(\cos (-k-t+4)-1) \theta (k+t-4)}{2 k}\text { if }k<-4$} \\
\end{align*}
✓ Sympy. Time used: 1.183 (sec). Leaf size: 201
from sympy import *
t = symbols("t")
k = symbols("k")
y = Function("y")
ode = Eq(y(t) + Derivative(y(t), (t, 2)) - (-Heaviside(-k + t - 4) + Heaviside(k + t - 4))/(2*k),0)
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0}
dsolve(ode,func=y(t),ics=ics)
\[
y{\left (t \right )} = \left (\frac {\cos {\left (k - 4 \right )} \theta \left (k - 4\right )}{2 k} - \frac {\cos {\left (k + 4 \right )} \theta \left (- k - 4\right )}{2 k} + \frac {\theta \left (- k - 4\right )}{2 k} - \frac {\theta \left (k - 4\right )}{2 k}\right ) \cos {\left (t \right )} + \left (- \frac {\sin {\left (k - 4 \right )} \theta \left (k - 4\right )}{2 k} - \frac {\sin {\left (k + 4 \right )} \theta \left (- k - 4\right )}{2 k} + \frac {\cos {\left (k - 4 \right )} \delta \left (k - 4\right )}{2 k} - \frac {\cos {\left (k + 4 \right )} \delta \left (k + 4\right )}{2 k} - \frac {\delta \left (k - 4\right )}{2 k} + \frac {\delta \left (k + 4\right )}{2 k}\right ) \sin {\left (t \right )} + \frac {\cos {\left (k - t + 4 \right )} \theta \left (- k + t - 4\right )}{2 k} - \frac {\cos {\left (k + t - 4 \right )} \theta \left (k + t - 4\right )}{2 k} - \frac {\theta \left (- k + t - 4\right )}{2 k} + \frac {\theta \left (k + t - 4\right )}{2 k}
\]