Math Problem Statement
Find the Laplace transform of the given function: f(t) = { 0, t < 5; t^2 - 10t + 50, t ≥ 5 }
Solution
To find the Laplace transform of the given piecewise function , where:
0, & t < 5 \\ t^2 - 10t + 50, & t \geq 5 \end{cases}$$ we can proceed by expressing $$f(t)$$ in terms of the Heaviside (unit step) function, $$u(t - a)$$, which activates the function at $$t = 5$$. 1. **Rewrite the function using the Heaviside function:** $$f(t) = (t^2 - 10t + 50) \cdot u(t - 5)$$ 2. **Apply the Laplace Transform properties:** The Laplace transform of $$f(t) = g(t) \cdot u(t - a)$$ is given by: $$\mathcal{L}\{f(t)\} = e^{-as} \mathcal{L}\{g(t + a)\}$$ where $$g(t) = t^2 - 10t + 50$$ and $$a = 5$$. 3. **Shift the function:** Substitute $$t + 5$$ for $$t$$ in $$g(t)$$: $$g(t + 5) = (t + 5)^2 - 10(t + 5) + 50$$ Expanding this, we get: $$g(t + 5) = t^2 + 10t + 25 - 10t - 50 + 50 = t^2$$ 4. **Take the Laplace Transform of $$g(t + 5) = t^2$$:** The Laplace Transform of $$t^2$$ is $$\frac{2}{s^3}$$. 5. **Combine results:** Now we apply the shifting factor $$e^{-5s}$$: $$\mathcal{L}\{f(t)\} = e^{-5s} \cdot \frac{2}{s^3}$$ So, the Laplace transform of $$f(t)$$ is: $$\mathcal{L}\{f(t)\} = \frac{2e^{-5s}}{s^3}$$ Would you like more details on any of these steps, or have any questions? --- Here are 5 related questions you might find helpful: 1. How does the Heaviside function help in simplifying piecewise functions for the Laplace transform? 2. What are other common properties of the Laplace transform, like the differentiation and integration properties? 3. How would we approach finding the inverse Laplace transform of a function involving $$e^{-as}$$? 4. What are some applications of the Laplace transform in solving differential equations? 5. How does the Laplace transform handle discontinuities in functions? **Tip:** When dealing with piecewise functions, rewriting them with the Heaviside function often simplifies the Laplace transformation process.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Laplace Transform
Piecewise Functions
Heaviside Function
Formulas
Laplace transform of f(t) = g(t) * u(t - a) is e^{-as} * L{g(t + a)}
Laplace Transform of t^n = n!/s^{n+1}
Theorems
Heaviside (Unit Step) Function Theorem
Suitable Grade Level
Undergraduate (Calculus and Differential Equations)
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