Piecewise continuous - step function

In summary, the correct solution for taking the Laplace transform of the given differential equation and solving for Y(s) is (9/s^2)(1-e^-2s) / (s^2 + 4).
  • #1
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Homework Statement


Consider the following initial value problem:

y''+4y = 9t, 0<=t<2
...0, t>=2

Find the equation you get by taking the Laplace transform of the differential equation and solve for Y(s)

Homework Equations


The Attempt at a Solution


I am just having trouble with the step function:

Say g(t)= 9t, 0<=t<2
...0, t>=2

g(t)= (1-u(t-2))9t ?
= 9t-u(t-2)9t

So, Laplace{g(t)} = (9-9e-2s)/(s2)

SO, s2Y(s) + 4Y(s) = (9-9e-2s)/(s2)

And, Y(s) = (9-9e-2s)/((s2)(s2+4))

There is something wrong with this. If anyone could help me out that would be much appreciated! Thanks!
 
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  • #2


Your attempt at solving the problem is mostly correct, but there are a few small errors. First, the step function should be written as u(t-2), not u(t-2)9t. Also, the Laplace transform of 9t should be 9/s^2, not 9/s. Additionally, when factoring out the common factor of 9, the numerator should be (1-e^-2s), not (9-9e^-2s). So the correct solution would be:

s^2Y(s) + 4Y(s) = (9/s^2) - (9/s^2)e^-2s

Y(s) = (9/s^2)(1-e^-2s) / (s^2 + 4)

Hope this helps!
 

Related to Piecewise continuous - step function

1. What is a piecewise continuous function?

A piecewise continuous function is a mathematical function that is defined using different formulas on different intervals. This means that the function may have different rules or equations for different parts of its domain.

2. What is a step function?

A step function is a special type of piecewise continuous function that has a constant value within each interval and changes abruptly at the boundaries between intervals. This creates a "step-like" graph.

3. How is a step function different from a continuous function?

Unlike continuous functions, step functions have distinct values at specific points and are not defined for all real numbers. This means that step functions have discontinuities, or breaks, in their graphs.

4. What are some real-world applications of piecewise continuous - step functions?

Piecewise continuous - step functions are commonly used in economics, physics, and engineering to model situations where there are abrupt changes or sudden shifts in data. For example, they can be used to represent the price changes of a product, the movement of a particle, or the voltage output of a circuit.

5. How do you graph a piecewise continuous - step function?

To graph a piecewise continuous - step function, you must first identify the different intervals and equations that make up the function. Then, plot the points for each interval and connect them with a line or curve, depending on the type of discontinuity. Make sure to label any breaks or discontinuities on the graph.

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