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## Q. 2.6.3

Impulse Response of a Simple First-Order Model

Obtain the unit-impulse response of the following model in two ways: (a) by separation of variables and (b) with the Laplace transform. The initial condition is x(0−) = 3. What is the value of x(0+)?

$\overset{.}{x}=\delta (t)$

## Verified Solution

a. Integrate both sides of the equation to obtain

$\int_{x(0-)}^{x(t)} dx = \int_{0-}^{t} \delta (t) dt = \int_{0-}^{0+} \delta (t) dt + \int_{0+}^{t} \delta (t) dt = 1+0$

because the area under a unit impulse is 1. This gives

$x(t)=x(0-)=1 \quad \text{or} \quad x(t)=x(0-)+1=3+1=4$

This is the solution for t > 0 but not for t = 0−. Thus, x(0+) = 4 but x(0−) = 3, so the impulse has changed x(t) instantaneously from 3 to 4.

b. The transformed equation is

$sX(s)-x(0-)=1 \quad \text{or} \quad X(s)=\frac{1+x(0-)}{s}$

which gives the solution x(t) = 1 + x(0−) = 4. Note that the initial value used with the derivative property is the value of x at t = 0−

The initial-value theorem gives

$x(0+)=\underset{s \rightarrow \infty}{\text{lim}} s X(s)= \underset{s \rightarrow \infty}{\text{lim}} s \frac{1+x(0-)}{s}=1+3=4$

which is correct.