Math Problem Statement
can you solve this and give the answer
Solution
To solve the problem using the Laplace transform, let’s go through the steps:
Step 1: Apply the Laplace Transform
The differential equation given is: where is defined as:
\sin(\pi t), & 0 \leq t < 1 \\ 0, & t \geq 1 \end{cases}$$ with initial conditions $$y(0) = 0$$ and $$y'(0) = 0$$. ### Step 2: Take the Laplace Transform of Both Sides Using the Laplace transform properties: $$\mathcal{L}\{y''\} = s^2 Y(s) - s y(0) - y'(0)$$ $$\mathcal{L}\{4y\} = 4 Y(s)$$ Substitute the initial conditions $$y(0) = 0$$ and $$y'(0) = 0$$: $$s^2 Y(s) + 4 Y(s) = \mathcal{L}\{f(t)\}$$ So, we have: $$(s^2 + 4) Y(s) = \mathcal{L}\{f(t)\}$$ ### Step 3: Compute $$\mathcal{L}\{f(t)\}$$ Since $$f(t)$$ is a piecewise function, we can use the Laplace transform for piecewise functions. For $$f(t) = \sin(\pi t)$$ for $$0 \leq t < 1$$ and 0 otherwise, we can write it as: $$f(t) = \sin(\pi t) \cdot u(t) - \sin(\pi t) \cdot u(t - 1)$$ where $$u(t)$$ is the Heaviside (step) function. Then, $$\mathcal{L}\{f(t)\} = \mathcal{L}\{\sin(\pi t)\} - \mathcal{L}\{\sin(\pi t) u(t - 1)\}$$ The Laplace transform of $$\sin(\pi t)$$ is: $$\mathcal{L}\{\sin(\pi t)\} = \frac{\pi}{s^2 + \pi^2}$$ For $$\sin(\pi t) u(t - 1)$$, we use the time-shifting property: $$\mathcal{L}\{\sin(\pi (t - 1))\} = e^{-s} \cdot \frac{\pi}{s^2 + \pi^2}$$ Thus: $$\mathcal{L}\{f(t)\} = \frac{\pi}{s^2 + \pi^2} - e^{-s} \cdot \frac{\pi}{s^2 + \pi^2} = \frac{\pi (1 - e^{-s})}{s^2 + \pi^2}$$ ### Step 4: Substitute and Solve for $$Y(s)$$ Now, substitute $$\mathcal{L}\{f(t)\}$$ into the equation: $$(s^2 + 4) Y(s) = \frac{\pi (1 - e^{-s})}{s^2 + \pi^2}$$ Thus: $$Y(s) = \frac{\pi (1 - e^{-s})}{(s^2 + 4)(s^2 + \pi^2)}$$ This expression represents $$Y(s)$$, the Laplace transform of the solution $$y(t)$$. --- Would you like a step-by-step explanation on inverse Laplace transformation, or do you have specific questions about this solution? ### Related Questions to Explore Further: 1. How do you apply the Laplace transform to other types of piecewise functions? 2. What is the inverse Laplace transform of $$\frac{\pi (1 - e^{-s})}{(s^2 + 4)(s^2 + \pi^2)}$$? 3. How can the Heaviside function simplify the Laplace transformation of piecewise functions? 4. Can this method be applied to differential equations with different boundary conditions? 5. How does the time-shifting property affect the Laplace transform? ### Tip: When handling piecewise functions in Laplace transformations, always consider using the Heaviside function, as it simplifies applying time shifts in the function.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Laplace Transform
Differential Equations
Piecewise Functions
Formulas
Laplace transform of y'' is s^2 Y(s) - s y(0) - y'(0)
Laplace transform of sin(at) is a / (s^2 + a^2)
Time-shifting property: L{f(t - a) u(t - a)} = e^{-as} F(s)
Theorems
Time-shifting property of Laplace Transform
Laplace Transform for piecewise functions
Suitable Grade Level
Undergraduate Mathematics
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