Math Problem Statement
Solution
To find the Laplace Transform , we need to work with the piecewise function provided:
\begin{cases} 0, & \text{if } 0 < t < 1, \\ t, & \text{if } 1 \leq t < 2, \\ 0, & \text{if } t \geq 2. \end{cases}$$ The Laplace transform is defined as: $$L\{F(t)\} = \int_0^\infty F(t)e^{-st} dt.$$ Given the piecewise definition of $$F(t)$$, we will split the integral over the intervals where $$F(t)$$ is non-zero. Specifically, $$F(t) = t$$ for $$1 \leq t < 2$$, so we compute: $$L\{F(t)\} = \int_1^2 t e^{-st} dt.$$ ### Step 1: Compute $$\int_1^2 t e^{-st} dt$$ The integral involves the product of $$t$$ and $$e^{-st}$$, so we use integration by parts: $$\int t e^{-st} dt = \frac{-t e^{-st}}{s} + \frac{1}{s} \int e^{-st} dt.$$ The second integral is: $$\int e^{-st} dt = \frac{-e^{-st}}{s}.$$ Thus: $$\int t e^{-st} dt = \frac{-t e^{-st}}{s} - \frac{e^{-st}}{s^2}.$$ ### Step 2: Evaluate over the limits $$t = 1$$ to $$t = 2$$ Substitute this back and compute: $$\int_1^2 t e^{-st} dt = \left[ \frac{-t e^{-st}}{s} - \frac{e^{-st}}{s^2} \right]_1^2.$$ First, evaluate at $$t = 2$$: $$\frac{-2 e^{-2s}}{s} - \frac{e^{-2s}}{s^2}.$$ Then, evaluate at $$t = 1$$: $$\frac{-1 e^{-s}}{s} - \frac{e^{-s}}{s^2}.$$ Combine these results: $$\int_1^2 t e^{-st} dt = \left( \frac{-2 e^{-2s}}{s} - \frac{e^{-2s}}{s^2} \right) - \left( \frac{-e^{-s}}{s} - \frac{e^{-s}}{s^2} \right).$$ Simplify: $$\int_1^2 t e^{-st} dt = \frac{-2 e^{-2s}}{s} - \frac{e^{-2s}}{s^2} + \frac{e^{-s}}{s} + \frac{e^{-s}}{s^2}.$$ Group terms: $$\int_1^2 t e^{-st} dt = \frac{e^{-s} - 2 e^{-2s}}{s} + \frac{e^{-s} - e^{-2s}}{s^2}.$$ ### Final Answer: The Laplace Transform is: $$L\{F(t)\} = \frac{e^{-s} - 2 e^{-2s}}{s} + \frac{e^{-s} - e^{-2s}}{s^2}.$$ --- Would you like a more detailed explanation of any specific step or a numerical example? Let me know! ### Related Questions: 1. What is the Laplace transform of other piecewise functions? 2. How is the method of integration by parts applied in Laplace transforms? 3. How do step functions ($$u(t)$$) simplify piecewise Laplace transforms? 4. What are the properties of Laplace transforms for discontinuous functions? 5. How can Laplace transforms be used in solving differential equations? ### Tip: When working with piecewise functions, consider rewriting them using unit step (Heaviside) functions to simplify integration and calculation of Laplace transforms.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Laplace Transform
Piecewise Functions
Integration by Parts
Formulas
Laplace Transform: L{f(t)} = ∫₀^∞ f(t)e^{-st} dt
Integration by Parts: ∫u dv = uv - ∫v du
Theorems
Laplace Transform of a piecewise function is computed by splitting the integral over intervals where the function is non-zero.
Suitable Grade Level
College/University Level
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