Find the Laplace transform of the piecewise defined functions f(t) (illustrated below) by expressing the functions in terms of the piecewise function and the Heaviside step function, H(t). (a) Find L[f(t)]. Assume that 0 < a=””>< c=”” always=”” holds.=”” f(t)=”” (b,=”” 2)=”” (c,=”” 2)=”” (a,=”” 0)=”” (c,=”” 0)=”” t=”” none=””>
The Correct Answer and Explanation is :
To solve for the Laplace transform of the piecewise function given, we need to express it in terms of Heaviside step functions ( H(t) ) to handle the transitions in the piecewise definition. However, the link you provided does not display the actual graph or the piecewise function. Based on your description, I will guide you through the general process of solving such a problem.
Steps for Solving the Laplace Transform of a Piecewise Function
- Understanding Piecewise Function: A piecewise function is typically defined for different intervals of ( t ). We need to express each interval using the Heaviside step function, ( H(t – t_0) ), where ( t_0 ) is the time at which the function starts to change. The Heaviside function is 0 before ( t = t_0 ) and 1 after.
- Rewriting the Function Using Heaviside:
For example, suppose we have a function ( f(t) ) as:
[
f(t) =
\begin{cases}
0 & \text{for } t < a \
c & \text{for } a \leq t < b \
b & \text{for } t \geq c
\end{cases}
]
We would rewrite it using Heaviside functions as:
[
f(t) = c \left[ H(t – a) – H(t – b) \right] + b H(t – c)
] - Laplace Transform of Each Term:
- The Laplace transform of ( H(t – t_0) ) is ( \frac{e^{-t_0 s}}{s} ).
- The Laplace transform of ( f(t) = A \cdot H(t – t_0) ) is ( \frac{A e^{-t_0 s}}{s} ).
- Taking the Laplace Transform: Now, for each part of the piecewise function, apply the Laplace transform. For example:
- The Laplace transform of ( c \left[ H(t – a) – H(t – b) \right] ) is:
[
c \left( \frac{e^{-a s}}{s} – \frac{e^{-b s}}{s} \right)
] - The Laplace transform of ( b H(t – c) ) is:
[
\frac{b e^{-c s}}{s}
]
- Final Laplace Transform: Adding all parts together gives the overall Laplace transform of ( f(t) ).
Example of a Full Solution
If ( f(t) ) were defined as follows:
[
f(t) =
\begin{cases}
0 & \text{for } t < 2 \
2 & \text{for } 2 \leq t < 4 \
5 & \text{for } t \geq 4
\end{cases}
]
We would rewrite it as:
[
f(t) = 2 \left[ H(t – 2) – H(t – 4) \right] + 5 H(t – 4)
]
Then, the Laplace transform is:
[
\mathcal{L}[f(t)] = 2 \left( \frac{e^{-2s}}{s} – \frac{e^{-4s}}{s} \right) + \frac{5 e^{-4s}}{s}
]
This process involves breaking the function into parts, applying the Laplace transform to each, and then combining the results.
If you have more specific details about the function or its intervals, feel free to share, and I can refine the solution further.