**2.1** The periodic function *f(t)* has the line spectrum shown in Figure P2.1. Assuming that *f(t)* is an even function, determine the following:

**(a)** The trigonometric Fourier series for *f(t)*.

**(b)** The complex Fourier series for *f(t)*.

**2.2.** Find *v(t)* in the circuit of Figure P2.2 using the Laplace transform.

**2.3.** Find *x(t)* for *t*0 for

*d*^{2}*x*(*t*)/*dt*^{2} + *x*(*t*) = 0

where *x*(0) = 1 and *dx*(0)/*dt* = −1.

**2.4.** Consider the Laplace transform

Using the final-value theorem, determine *f*(∞). Check your answer by finding the inverse Laplace transform *f(t)* and letting *t* → ∞.

**2.5.** The initial conditions for the following differential equation

are given by

**(a)** Write in its simplest form the Laplace transform of the function *Y(s)*, by taking the Laplace transform of this differential equation.

**(b)** Expand *Y(s)* by means of the partial fraction expansion method. Determine all unknown constants. ...

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