50. Solve the integral equation (0) f(t) = e' + e¹ e² + e[e-²f(7) dr. 14 to subooong Jo
50. Solve the integral equation (0) f(t) = e' + e¹ e² + e[e-²f(7) dr. 14 to subooong Jo
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Use the laplace transform to solve the given integral equation or intergordifferential equation
![**Title: Solving Integral Equations**
**Problem 50:**
**Objective:**
Solve the integral equation.
**Equation:**
\[ f(t) = e^t + e^t \int_0^t e^{-\tau} f(\tau) \, d\tau \]
**Explanation:**
This is an example of an integral equation where the function \( f(t) \) is defined in terms of an integral involving itself. The equation consists of two key terms on the right-hand side:
- The first term, \( e^t \), which is a straightforward exponential function.
- The second term involves an integration from \( 0 \) to \( t \) of the product \( e^{-\tau} f(\tau) \), which contributes a more complex relationship between \( f(t) \) and its integral over time.
To solve this equation, methods like substitution or differentiation under the integral sign might be employed, often starting with the identification or assumption of a possible form for \( f(t) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F776fde46-0e13-4f4b-965e-6a3678bd5960%2Fdb9cbb13-dc62-42a9-9aa9-28d7f96151be%2Fgs4bdch_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Solving Integral Equations**
**Problem 50:**
**Objective:**
Solve the integral equation.
**Equation:**
\[ f(t) = e^t + e^t \int_0^t e^{-\tau} f(\tau) \, d\tau \]
**Explanation:**
This is an example of an integral equation where the function \( f(t) \) is defined in terms of an integral involving itself. The equation consists of two key terms on the right-hand side:
- The first term, \( e^t \), which is a straightforward exponential function.
- The second term involves an integration from \( 0 \) to \( t \) of the product \( e^{-\tau} f(\tau) \), which contributes a more complex relationship between \( f(t) \) and its integral over time.
To solve this equation, methods like substitution or differentiation under the integral sign might be employed, often starting with the identification or assumption of a possible form for \( f(t) \).
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