How to Solve Initial Value Problem using Heaviside Functions

In summary, to solve the initial value problem with the given piecewise function, first solve for the case where 0 < k < 4, and then use that solution to find A and B in order to solve the problem for the general case.
  • #1
danj303
15
0

Homework Statement



Consider the initial value problem

y'' + 1/3y' + 4y = fk(t)

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

fk(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 fk(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!
 
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  • #2
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.
 
  • #3
That makes sence. So how do I go from that to solving the initial value problem??
 
  • #4
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.
 

1. What is a Heaviside function?

A Heaviside function, also known as the unit step function, is a mathematical function that returns 0 for negative inputs and 1 for non-negative inputs. It is commonly denoted as H(x) or u(x).

2. What is the purpose of a Heaviside function?

Heaviside functions are often used in mathematics and engineering to represent a discontinuity or a sudden change in a function. They are also useful in representing certain physical phenomena, such as switching on and off of a current or signal.

3. How is a Heaviside function defined mathematically?

The Heaviside function can be defined mathematically as follows:

H(x) = { 0, if x < 0
1, if x ≥ 0 }

4. What are the properties of a Heaviside function?

Some important properties of Heaviside functions include:

  • H(x) is an even function, i.e. H(-x) = H(x)
  • H(x) is continuous for all real values of x
  • The derivative of H(x) is the Dirac delta function, denoted as δ(x)
  • H(x) is often used in combination with other functions to represent piecewise defined functions

5. How is a Heaviside function used in solving differential equations?

Heaviside functions are commonly used in solving differential equations, particularly in the method of Laplace transforms. They can help simplify the solution by representing a discontinuity or a sudden change in the function being solved for. They are also useful in boundary value problems and initial value problems.

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