Math Problem Statement
Solution
To solve the Laplace transform of the given piecewise function , we can break the process into steps.
The function is defined as:
\begin{cases} 0, & t < 6 \\ (t - 6)^5, & t \geq 6 \end{cases}$$ ### Step 1: Use the Heaviside (Unit Step) Function We can express the function in terms of the Heaviside step function $$u(t - 6)$$. The given function can be written as: $$f(t) = (t - 6)^5 u(t - 6)$$ ### Step 2: Laplace Transform of Shifted Function The Laplace transform of a shifted function $$(t - a)^n u(t - a)$$ is given by the formula: $$\mathcal{L}\{(t - a)^n u(t - a)\} = \frac{n!}{s^{n+1}} e^{-as}$$ where $$a = 6$$ and $$n = 5$$. ### Step 3: Apply the Formula Substitute the values $$n = 5$$ and $$a = 6$$ into the formula: $$\mathcal{L}\{(t - 6)^5 u(t - 6)\} = \frac{5!}{s^{5+1}} e^{-6s}$$ Simplify $$5!$$: $$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$$ Thus, the Laplace transform is: $$\mathcal{L}\{f(t)\} = \frac{120}{s^6} e^{-6s}$$ ### Final Answer: $$\mathcal{L}\{f(t)\} = \frac{120 e^{-6s}}{s^6}$$ Would you like further details or have any specific questions about this solution? Here are 5 related questions for further exploration: 1. How do we apply the Heaviside function in more complex Laplace transforms? 2. What is the Laplace transform of $$(t - a)^n u(t - a)$$ for general $$n$$ and $$a$$? 3. How can Laplace transforms handle more intricate piecewise functions? 4. Can you explain the role of shifting the function by a constant $$a$$ in Laplace transforms? 5. How does the Laplace transform change when dealing with discontinuous functions? **Tip:** When using the Heaviside function in Laplace transforms, it helps to carefully identify the shift in time to avoid mistakes in applying the exponential term $$e^{-as}$$.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Laplace Transform
Heaviside Step Function
Piecewise Functions
Formulas
Laplace Transform of (t - a)^n u(t - a): L{(t - a)^n u(t - a)} = n!/s^{n+1} e^{-as}
Theorems
Laplace Transform Theorem for Shifted Functions
Factorial Calculation for Laplace Transforms
Suitable Grade Level
College/Advanced Calculus
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