Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
![**Problem Statement:**
Use Laplace transform to solve for \( f(t) \):
\[ f(t) + \int_{0}^{t} (t - \tau) f(\tau) d\tau = t. \]
**Explanation:**
The equation involves solving for the function \( f(t) \) using the Laplace transform. The equation comprises a function \( f(t) \) added to an integral term, where the integral is calculated from 0 to \( t \) and involves the product of \( (t - \tau) \) and \( f(\tau) \). This is set equal to the variable \( t \).
The goal is to apply the Laplace transform to convert the differential or integral equations into algebraic equations, making them easier to solve. Once solved in the transformed domain, the inverse Laplace transform is used to convert the solution back to the time domain.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff26fda3f-4323-495f-b28b-f131dcfa5992%2F4a1f72a5-8eea-4f1d-99b2-092d564f10fe%2F31k5mnj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Use Laplace transform to solve for \( f(t) \):
\[ f(t) + \int_{0}^{t} (t - \tau) f(\tau) d\tau = t. \]
**Explanation:**
The equation involves solving for the function \( f(t) \) using the Laplace transform. The equation comprises a function \( f(t) \) added to an integral term, where the integral is calculated from 0 to \( t \) and involves the product of \( (t - \tau) \) and \( f(\tau) \). This is set equal to the variable \( t \).
The goal is to apply the Laplace transform to convert the differential or integral equations into algebraic equations, making them easier to solve. Once solved in the transformed domain, the inverse Laplace transform is used to convert the solution back to the time domain.
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