Question

Apply Inverse Laplace transform and obtain y(t) if a step input of amplitude 10 is applied to the system represented by the following differential equation. Assume zero initial conditions. y+10y+6y=7u(t)

Answer 1

To find y(t), we apply the Inverse Laplace transform to the differential equation after taking the Laplace transform and solving for the coefficients of the decomposed expression. The result, obtained by inverse transforming each term, gives the solution in the time domain.

Step-by-step explanation:

To apply the Inverse Laplace transform to obtain y(t) when a step input of amplitude 10 is applied to a system represented by the differential equation y+10y+6y=7u(t), we first take the Laplace transform of both sides of the equation, considering zero initial conditions:

L{y(t)} + 10L{y(t)} + 6L{y(t)} = 7L{u(t)}

Now, because L{u(t)} = 1/s for a unit step function, and the input is 10 times that, we get:

L{y(t)}(s^2 + 10s + 6) = 70/s

Next, we'll find the roots of the quadratic equation s^2 + 10s + 6:

The roots are s=-1 and s=-6. Therefore, we can decompose the above expression:

L{y(t)} = \frac{70}{(s+1)(s+6)} = \frac{A}{s+1}+\frac{B}{s+6}

By solving for coefficients A and B, we can then apply the Inverse Laplace transform to find y(t).

Let A(s+6) + B(s+1) = 70, and by setting s=-1 and s=-6 respectively, find A and B. The Inverse Laplace transform of the result gives the solution for y(t) in the time domain.