Question

+ 2y = 4u, y(0) = 0, for the following input: Solve: dt 0<t<T u(t) t>T Graph the solution (you may use Excel or Matlab) for T= 1sec, 0.1sec, 0.01sec, and 0.001sec. Do you see what is happening to the output? What is happening to the input?!

Answer 1

Solve   
49 + 2y = 4u, y(0) = 0,
where
So<t<T u(t) = { Tot>T
.

Step 1:

Find the solution for the homogeneous equation

49 + 2y = 0, y(0) = 0

$\frac{d y}{d t}=-2 y \Rightarrow \frac{d y}{y}=-2dt$

By integrating, we obtain  

-2 / dt = lny = -2t +O

ye = Coe- Core
, Now to get $C_0$ , we have
g(0) = 0 = Cu = 0 = 4c = 0.

Step 2:

Find a particular solution .

yple) = c+21* *Au(s)ds = { $(1-4-2) Ostst t>T

Then the general solution is given by
y = yc + Yp = (1-e-2) 0<t<T t>T 0
.

Applications Mon, Jan 27 1131 AM Figure 1 Window HAID File Edit View Insert Tools Desirop Dada DERE Solution for T=1

Applications Mon, Jan 27 11:32 AM Figure 1 Window HAID File Edit View Insert Tools Desirop Dada DERE Solution for T-0.1

Applications Mon, Jan 27 1133 AM Figure 1 Window HAID File Edit View Insert Tools Desinop Dada DERE Soluson for T=0.01

Applications Mon, Jan 27 1133 AM Figure 1 Window HAID File Edit View Insert Tools Desirop Dada DERE Soluson for T-0.002
The solution is diminishing (becomes zero function) as T is decreasing from 1 to 0.1 to 0.01 to 0.001.