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
True or false
![**Problem Statement:**
Show that the function:
\[ y = -x \cos(x) - x \]
is a solution to the differential equation:
\[ x \left(\frac{dy}{dx}\right) = y + x^2 \sin(x). \]
---
To demonstrate that the function \( y = -x \cos(x) - x \) satisfies the given differential equation, follow these steps:
1. **Determine \( \frac{dy}{dx} \)**: Differentiate the given function \( y = -x \cos(x) - x \) with respect to \( x \).
2. **Substitute into the Differential Equation**: Substitute \( y \) and \( \frac{dy}{dx} \) into the differential equation \( x \frac{dy}{dx} = y + x^2 \sin(x) \).
3. **Verify the Equality**: Show that both sides of the equation are equal after substitution.
This process will confirm that the provided function is indeed a solution to the differential equation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4d5fe7c4-bc88-41ab-8c75-523dd8842f64%2F3c4feead-b211-4ed8-b537-3a54a3298f81%2Fd489gzf_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Show that the function:
\[ y = -x \cos(x) - x \]
is a solution to the differential equation:
\[ x \left(\frac{dy}{dx}\right) = y + x^2 \sin(x). \]
---
To demonstrate that the function \( y = -x \cos(x) - x \) satisfies the given differential equation, follow these steps:
1. **Determine \( \frac{dy}{dx} \)**: Differentiate the given function \( y = -x \cos(x) - x \) with respect to \( x \).
2. **Substitute into the Differential Equation**: Substitute \( y \) and \( \frac{dy}{dx} \) into the differential equation \( x \frac{dy}{dx} = y + x^2 \sin(x) \).
3. **Verify the Equality**: Show that both sides of the equation are equal after substitution.
This process will confirm that the provided function is indeed a solution to the differential equation.
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