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Use the Laplace transform to solve the given integral equation. f'(t)=1-sin(t)- ff(r)dr_f(0) = 0 f(t) = sin(t) — -— -tsin(t) b. t²e¹ + t²e¹ 3te¹ 4 4 c. f(t)=sin(t) + cos(t) a. f(t) = - 4t 64sin(√17t) 17√17 27sin (√10t) 10√10 d. f(t)= + 17 e-t 8 3t e. f(t)= + 10

Use the Laplace transform to solve the given integral equation. f'(t)=1-sin(t)- ff(r)dr_f(0) = 0 f(t) = sin(t) — -— -tsin(t) b. t²e¹ + t²e¹ 3te¹ 4 4 c. f(t)=sin(t) + cos(t) a. f(t) = - 4t 64sin(√17t) 17√17 27sin (√10t) 10√10 d. f(t)= + 17 e-t 8 3t e. f(t)= + 10

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter9: Multivariable Calculus
Section9.CR: Chapter 9 Review
Problem 54CR
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**Using the Laplace Transform to Solve the Given Integral Equation** Given integral equation: \[ f'(t) = 1 - \sin(t) - \int_0^t f(\tau) d\tau \quad \text{with} \quad f(0) = 0 \] Possible solutions for \( f(t) \): a. \( f(t) = \sin(t) - \frac{1}{2} t \sin(t) \) b. \( f(t) = \frac{t^2 e^t}{4} + \frac{3 t e^t}{4} - \frac{e^{-t}}{8} - \frac{e^t}{8} \) c. \( f(t) = \sin(t) + \cos(t) \) d. \( f(t) = \frac{4t}{17} + \frac{64 \sin(\sqrt{17} t)}{17 \sqrt{17}} \) e. \( f(t) = \frac{3t}{10} + \frac{27 \sin(\sqrt{10} t)}{10 \sqrt{10}} \) To solve this problem using the Laplace transform, follow these general steps: 1. Take the Laplace transform of both sides of the given equation. 2. Use the properties of the Laplace transform to handle the derivative and the integral. 3. Solve the resulting algebraic equation in the Laplace domain. 4. Take the inverse Laplace transform to find \( f(t) \). Using the Laplace transform can simplify the process of solving differential and integral equations by converting them into algebraic problems. Each proposed solution must be checked to ensure it satisfies both the differential equation and the initial condition.
Transcribed Image Text:**Using the Laplace Transform to Solve the Given Integral Equation** Given integral equation: \[ f'(t) = 1 - \sin(t) - \int_0^t f(\tau) d\tau \quad \text{with} \quad f(0) = 0 \] Possible solutions for \( f(t) \): a. \( f(t) = \sin(t) - \frac{1}{2} t \sin(t) \) b. \( f(t) = \frac{t^2 e^t}{4} + \frac{3 t e^t}{4} - \frac{e^{-t}}{8} - \frac{e^t}{8} \) c. \( f(t) = \sin(t) + \cos(t) \) d. \( f(t) = \frac{4t}{17} + \frac{64 \sin(\sqrt{17} t)}{17 \sqrt{17}} \) e. \( f(t) = \frac{3t}{10} + \frac{27 \sin(\sqrt{10} t)}{10 \sqrt{10}} \) To solve this problem using the Laplace transform, follow these general steps: 1. Take the Laplace transform of both sides of the given equation. 2. Use the properties of the Laplace transform to handle the derivative and the integral. 3. Solve the resulting algebraic equation in the Laplace domain. 4. Take the inverse Laplace transform to find \( f(t) \). Using the Laplace transform can simplify the process of solving differential and integral equations by converting them into algebraic problems. Each proposed solution must be checked to ensure it satisfies both the differential equation and the initial condition.
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