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Find f(x) if y = f(x) satisfies dy dx e¯½ f sin x and ƒ (7) = ²/

Find f(x) if y = f(x) satisfies dy dx e¯½ f sin x and ƒ (7) = ²/

Calculus: Early Transcendentals
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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Please solve the following calc question (use calc 2 concepts) 

**Problem Statement:** Find \( f(x) \) if \( y = f(x) \) satisfies \[ \frac{dy}{dx} = e^{-y} \sin x \] and \[ f\left(\frac{\pi}{2}\right) = \frac{1}{5} \] --- **Explanation:** In this problem, we are looking for a function \( f(x) \) given that it satisfies the differential equation \[ \frac{dy}{dx} = e^{-y} \sin x \] with an initial condition \[ f\left(\frac{\pi}{2}\right) = \frac{1}{5} \] This is a first-order differential equation. Solving it involves integrating factors or separation of variables to find the specific function \( f(x) \) that meets the given condition.
Transcribed Image Text:**Problem Statement:** Find \( f(x) \) if \( y = f(x) \) satisfies \[ \frac{dy}{dx} = e^{-y} \sin x \] and \[ f\left(\frac{\pi}{2}\right) = \frac{1}{5} \] --- **Explanation:** In this problem, we are looking for a function \( f(x) \) given that it satisfies the differential equation \[ \frac{dy}{dx} = e^{-y} \sin x \] with an initial condition \[ f\left(\frac{\pi}{2}\right) = \frac{1}{5} \] This is a first-order differential equation. Solving it involves integrating factors or separation of variables to find the specific function \( f(x) \) that meets the given condition.
Expert Solution
Step 1: We will analyze the given question

Given the experience fraction numerator d y over denominator d x end fraction equals e to the power of negative y end exponent sin x 

and y=f(x) 

we have to find the solution f(x) by the condition f left parenthesis straight pi over 2 right parenthesis equals 1 fifth

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