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High pass filter and signal

Hi,

I have this problem about signal that I have a little trouble to finish:

I attached a small figure of the circuit.

I have a High pass filter, and the equation is:

$\displaystyle \frac{dx[t]}{dt}=\frac{y[t]}{RC}+ \frac{dy[t]}{dt}$

the solution for the equation is:

$\displaystyle y[t]=\int_{-\infty}^{t} e^{-(t-\lambda)/\tau}x'[\lambda] \,d\lambda $

This is what I found and it matches with the answer from the professor.

But the question said:

**Find the solution of the ODE to obtain y[t] as some integral of **__x[t]__.

Now I need to take away the $\displaystyle x'[\lambda]$ and replace by $\displaystyle x[\lambda]$.

What i did is to use the integration by part to the solution $\displaystyle y[t]$.

I got:

$\displaystyle y[t]=x[t]+\lambda\int_{-\infty}^{t}x(\lambda)e^{-(t-\lambda)/\tau} \,d\lambda $

My problem is that by intergrating, even though I took away what I wanted, I have introduced the variable **t**. I dont know if it is a problem.

My second question is the following:

**Find the impulse response function $\displaystyle h[t]$ so that the solution is has the form:**

$\displaystyle y[t]=\int_{-\infty}^{+\infty}h(t-\lambda) x(\lambda) \,d\lambda $

Hint: if you find a lonely x(t), remenber that: $\displaystyle x[t]=\int_{-\infty}^{+\infty}\delta(t-\lambda) x(\lambda) \,d\lambda $

Here I am confused with the boundaries of the integration

Please can I have some help with the problem

Thank you