How to Solve Initial Value Problem using Heaviside Functions

[danj303]

Homework Statement

Consider the initial value problem

y'' + ¹/₃y' + 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!

 

[betel]

If you plot the two different combinations of the Heaviside functions you will notice, that both are identical.
The one from Maple just has an additional redundant factor.

 

[danj303]

That makes sence. So how do I go from that to solving the initial value problem??

 

[HallsofIvy]

The simplest way is to do it as two separate problems.

For 0< k< 4, 0< 4-k < k so first solve
y'' + (1/3)y' + 4y = 1/(2k), y(0)= y'(0)= 0

Use that solution to find A= y(k) and B= y'(k)

Then solve
y"+ (1/3)y'+ 4y= 0, y(k)= A, y'(k)= B.