Question 2.7.4: Exponential Response of a First-Order Model Use the Laplace ...

Taking the transform of both sides of the equation, we obtain

sX(s)-x(0)+5X(s)=\frac{7}{(s+3)^2}

Solve for X(s) using the given value of x(0).

X(s)=\frac{7}{(s+3)^2(s+5)}

The partial-fraction expansion was obtained in Example 2.7.3. It is

X(s)=\frac{7}{2(s+3)^2}-\frac{7}{4(s+3)}+\frac{7}{4(s+5)}

and the inverse transform is

x(t)=\frac{7}{2}te^{-3t}-\frac{7}{4}e^{-3t}+\frac{7}{4}e^{-5t}

The plot of the response is shown in Figure 2.7.2. The “hump” in the response is caused by the multiplicative factor of t in the input 7te^{−3t}.