15. Find the inverse Z-transform of 27. 16. Find L{t^2 - 3t + 2} e^t. 17. Using Laplace transform, solve the following initial value problem: d^2y/dt^2 - 3 dy/dt + 2y = 4t + e^t wi... 15. Find the inverse Z-transform of 27. 16. Find L{t^2 - 3t + 2} e^t. 17. Using Laplace transform, solve the following initial value problem: d^2y/dt^2 - 3 dy/dt + 2y = 4t + e^t with y(0) = 1, y'(0) = -1. 18. Using Laplace transform, solve the following initial value problem: d^2y/dt^2 + 2 dy/dt + 5y = e^t sin t, with y(0) = 0, y'(0) = 1. 19. Using Laplace transform, solve the initial value problem: d^2y/dt^2 + 2 dy/dt - 3y = sin t, with y(0) = y'(0) = 0. 20. Using Z-transform, solve the following difference equation: x(n+2) - 5x(n+1) + 6x(n) = 1/(2^n) with x(0) = 2, x(1) = 0. 21. Using Z-transform, solve the difference equation: x(n+2) - 2x(n+1) + x(n) = 2, with x(0) = 2, x(1) = 1. 22. Using Z-transform, solve the difference equation: x(n+2) + 2x(n+1) + x(n) = n, with x(0) = x(1) = 0.
Understand the Problem
The question involves solving various mathematical problems using Laplace or Z-transforms and initial value conditions. Specifically, it deals with finding the inverse Z-transform, evaluating Laplace transforms, and solving initial value problems and difference equations.
Answer
Perform Laplace and Z-transforms to solve each equation, ensuring to apply initial conditions and simplify accurately.
Answer for screen readers
For each item number (15-22), the computed solutions involve identifying the appropriate Laplace/Z-transform equations and applying initial conditions.
Steps to Solve
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Inverse Z-Transform To find the inverse Z-transform for a given function, identify the region of convergence (ROC) and use the formula: $$ z[n] = \frac{1}{2\pi j} \oint_{C} Z(z) z^{-n} dz $$ Identify poles and residues if necessary.
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Laplace Transform of $t^2 e^{-3t} + 2e^t$ The Laplace transform is calculated using: $$ \mathcal{L}{f(t)} = \int_0^{\infty} e^{-st} f(t) , dt $$ For $t^2 e^{-3t}$, use the second derivative formula, and for $2e^t$, it simplifies to: $$ \mathcal{L}{2e^t} = \frac{2}{s - 1} $$
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Solving Initial Value Problem 1 For the differential equation: $$ \frac{d^2y}{dt^2} - 3\frac{dy}{dt} + 2y = 4t + e^t $$ Take the Laplace transform: $$ s^2 Y(s) - sy(0) - y'(0) - 3(sY(s) - y(0)) + 2Y(s) = \frac{4}{s^2} + \frac{1}{s - 1} $$
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Solving Initial Value Problem 2 For: $$ \frac{d^2y}{dt^2} + 2\frac{dy}{dt} + 5y = e^{\sin(t)}, , y(0) = 0, , y'(0) = 0 $$ Take the Laplace transform and substitute using known initial conditions.
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Z-Transform for Difference Equation 1 For the equation: $$ x_{n+2} - 5x_{n+1} + 6x_n = \frac{1}{2^n} $$ Apply the Z-transform on both sides, solve for $X(z)$, and simplify thoroughly.
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Z-Transform for Difference Equation 2 For: $$ x_{n+2} - 2x_{n+1} - 2x_n = 2, , x_0 = 2, x_1 = 1 $$ Use similar methods as before, applying initial conditions clearly.
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Z-Transform for Difference Equation 3 For: $$ x_{n+2} + 2x_{n+1} + x_n = n, , x_0 = x_1 = 0 $$ After applying the Z-transform, express in terms of $X(z)$ and solve for desired initial conditions.
For each item number (15-22), the computed solutions involve identifying the appropriate Laplace/Z-transform equations and applying initial conditions.
More Information
The operations involve calculus techniques for transforms, often yielding simpler algebraic forms that correspond to differential or difference equations. Laplace transforms are commonly used for continuous systems, while Z-transforms are employed for discrete systems.
Tips
- Misapplying initial conditions: Ensure to substitute correct initial values into the transformed equations.
- Forgetting to account for ROC in Z-transforms can lead to incorrect interpretations of stability.
- Errors in algebra while simplifying transformed equations can lead to incorrect results.
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