Question

Solve (find the impulse response) x ′′ +2x ′+x=δ(t),x(0)=0,x′(0)=0.

Answer 1

To find the impulse response for the given differential equation, we can solve it using the Laplace transform method.

Step-by-step explanation:

The given equation is a second-order linear ordinary differential equation with constant coefficients. To solve for the impulse response, we can use the Laplace transform method.

Applying the Laplace transform to both sides of the equation, we get:

s^2X(s) + 2sX(s) + X(s) = 1 (where X(s) is the Laplace transform of x(t))

Now, we can rearrange this equation and solve for X(s) in terms of the Laplace variable 's'. We inverse transform X(s) to get the impulse response x(t).