Use the Laplace transform to solve the given integral equation. of "si f(t)=3t-9 a. f(t)=sin(t)-tsin( b. f(t)= + 4 t²e¹ 3te¹ 4 c. f(t)=sin(t) + cos(t) d. f(t)= O a 3t e. f(t)= + 10 O b sin(T)f(t-T)dt -tsin(t) O C Od Oe e-t 8 4t 64sin(√17t) + 17 17√17 27sin(√10t) 10√10 8
Use the Laplace transform to solve the given integral equation. of "si f(t)=3t-9 a. f(t)=sin(t)-tsin( b. f(t)= + 4 t²e¹ 3te¹ 4 c. f(t)=sin(t) + cos(t) d. f(t)= O a 3t e. f(t)= + 10 O b sin(T)f(t-T)dt -tsin(t) O C Od Oe e-t 8 4t 64sin(√17t) + 17 17√17 27sin(√10t) 10√10 8
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Use the Laplace transform to solve the given integral equation.**
\[ f(t) = 3t - 9 \int_0^t \sin(\tau)f(t-\tau)d\tau \]
a. \( f(t) = \sin(t) - \frac{1}{2} t \sin(t) \)
b. \( f(t) = \frac{t^2 e^t}{4} + \frac{3t e^t}{4} - \frac{e^{-t}}{8} - \frac{e^t}{8} \)
c. \( f(t) = \sin(t) + \cos(t) \)
d. \( f(t) = \frac{4t}{17} + \frac{64 \sin(\sqrt{17} t)}{17\sqrt{17}} \)
e. \( f(t) = \frac{3t}{10} + \frac{27 \sin(\sqrt{10} t)}{10 \sqrt{10}} \)
### Multiple Choice Options
- ⃝ a
- ⃝ b
- ⃝ c
- ⃝ d
- ⃝ e
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This problem requires using the Laplace transform to solve a given integral equation. The equation provided is:
\[ f(t) = 3t - 9 \int_0^t \sin(\tau) f(t - \tau) d\tau \]
The options provided offer different functions \( f(t) \) as potential solutions. The correct solution must satisfy the given integral equation through a proper application of the Laplace transform. Examine each option closely, compute \( f(t) \), and verify which one meets the criteria set by the integral equation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa72e8965-bd79-4464-a740-a72095a0be2c%2Fede2dd46-e17d-4eb0-802d-1d55b3586cea%2Fbgwvo5q_processed.png&w=3840&q=75)
Transcribed Image Text:**Use the Laplace transform to solve the given integral equation.**
\[ f(t) = 3t - 9 \int_0^t \sin(\tau)f(t-\tau)d\tau \]
a. \( f(t) = \sin(t) - \frac{1}{2} t \sin(t) \)
b. \( f(t) = \frac{t^2 e^t}{4} + \frac{3t e^t}{4} - \frac{e^{-t}}{8} - \frac{e^t}{8} \)
c. \( f(t) = \sin(t) + \cos(t) \)
d. \( f(t) = \frac{4t}{17} + \frac{64 \sin(\sqrt{17} t)}{17\sqrt{17}} \)
e. \( f(t) = \frac{3t}{10} + \frac{27 \sin(\sqrt{10} t)}{10 \sqrt{10}} \)
### Multiple Choice Options
- ⃝ a
- ⃝ b
- ⃝ c
- ⃝ d
- ⃝ e
---
This problem requires using the Laplace transform to solve a given integral equation. The equation provided is:
\[ f(t) = 3t - 9 \int_0^t \sin(\tau) f(t - \tau) d\tau \]
The options provided offer different functions \( f(t) \) as potential solutions. The correct solution must satisfy the given integral equation through a proper application of the Laplace transform. Examine each option closely, compute \( f(t) \), and verify which one meets the criteria set by the integral equation.
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