Express T(x) in terms of the Heaviside function. Find the Laplace transform of T(x). Solve the differential equation y'' − y = T(x)

Question

The tent function is defined by T(x) = x 0 < x < 1/2

1 − x 1/2 ≤ x ≤ 1

0 otherwise

a) Express T(x) in terms of the Heaviside function.

(b) Find the Laplace transform of T(x).

(c) Solve the differential equation y'' − y = T(x), y(0) = y'(0) = 0

Answer 1

First of all, we 'turn on' the function x by simply writing f(x) = x + ...

Next, we turn off the function x at x = 1/2 by subtracting xH(x-1/2). So far, f(x) = x-xH(x-1/2)+...

Next, we turn on the function 1-x at x = 1/2 by adding (1-x)H(x-1/2). Now, f(x) = x-xH(x-1/2)+(1-x)H(x-1/2)+...

Finally, we turn off the function 1-x at x = 1 by subtracting (1-x)H(x-1). Thus,

f(x) = x-xH(x-1/2)+(1-x)H(x-1/2)-(1-x)H(x-1). Collecting like terms,

f(x) = x+(1-2x)H(x-1/2)-(1-x)H(x-1).