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## Homework Statement

Consider the initial value problem

y'' +

^{1}/

_{3}y' + 4y = f

_{k}(t)

with y(0) = y'(0) = 0,

f

_{k}(t) = piecewise function

^{1}/

_{2k}if 4 - k <= t < 4 + k

0 otherwise

and 0 < k < 4

(a) Sketch the graph of fk(t). Observe that the area under the graph is independent of k.

(b) Write fk(t) in terms of Heaviside step functions and then solve the initial value problem.

(c) Plot the solution for k = 2, k = 1 and k = 1/2. Describe how the solution depends on k.

## The Attempt at a Solution

I can sketch the graph fine etc, just struggling with putting f

_{k}(t) in terms of a heaviside function and then solving the initial value problem

I tried on maple and it gave me a heaviside function

(1/2k)*Heaviside(t-4+k)-(1/2k)*Heaviside(t-4+k)*Heaviside(t-4-k)

But by my calculation it should be

(1/2k)*Heaviside(t-4+k)-(1/2k)*Heaviside(t-4-k)

Thanks for any help!!