Solve the initial value problem y′+y=f(t),y(0)=0 where f(t)={1,−1, if t<4 if t≥4 Use h(t−a) for the Heaviside function shifted a units horizontally.

Looks like the function on the right hand side is

We can write it in terms of the Heaviside function,

as

Now for the ODE: take the Laplace transform of both sides:

Solve for Y(s), then take the inverse transform to solve for y(t):

the answer is y < -3 or   y > 3

-3y > 7 + 2

-3 y > 9

y < -3