Math Problem Statement
evaluate the laplace transform of the function f(t)={0 t<3 (t-3)^3 t>=3
Solution
To evaluate the Laplace transform of the function , which is defined as
\begin{cases} 0, & t < 3 \\ (t - 3)^3, & t \geq 3 \end{cases}$$ we can approach this using the **Laplace transform for shifted functions**. ### Step 1: Recognize the Shift Since $$f(t) = (t - 3)^3$$ for $$t \geq 3$$, this suggests a **shifted function**. The Heaviside (unit step) function $$u(t - 3)$$ can be used to rewrite $$f(t)$$ as: $$f(t) = (t - 3)^3 u(t - 3)$$ ### Step 2: Use the Laplace Transform of a Shifted Function The Laplace transform of a function $$g(t - a) u(t - a)$$ is given by: $$\mathcal{L}\{g(t - a) u(t - a)\} = e^{-as} G(s)$$ where $$G(s)$$ is the Laplace transform of $$g(t)$$ (i.e., $$G(s) = \mathcal{L}\{g(t)\}$$). In this case, $$g(t) = t^3$$ and $$a = 3$$. ### Step 3: Find the Laplace Transform of $$g(t) = t^3$$ The Laplace transform of $$t^3$$ is: $$\mathcal{L}\{t^3\} = \frac{6}{s^4}$$ ### Step 4: Apply the Shift Now, using the shift property, we have: $$\mathcal{L}\{(t - 3)^3 u(t - 3)\} = e^{-3s} \cdot \frac{6}{s^4}$$ ### Final Answer Therefore, the Laplace transform of $$f(t)$$ is: $$\mathcal{L}\{f(t)\} = \frac{6 e^{-3s}}{s^4}$$ Would you like further details or have any questions? --- Here are some related questions you might find useful: 1. What is the Laplace transform of other shifted functions? 2. How does the Heaviside function simplify the transformation of piecewise functions? 3. How would this approach change if the exponent of $$t$$ was different? 4. What happens if the shift occurs at $$t = 0$$ rather than $$t = 3$$? 5. How would you take the inverse Laplace transform of similar expressions? **Tip:** Remember, the Heaviside (unit step) function is a useful tool for expressing and transforming piecewise functions in Laplace transforms.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Laplace Transform
Piecewise Functions
Shifted Functions
Formulas
L{g(t - a) u(t - a)} = e^{-as} G(s)
L{t^n} = n!/s^{n+1}
Theorems
Shift Theorem for Laplace Transforms
Suitable Grade Level
Grades 11-12
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