For the differential equation 3 y" + x*y" x² +1 Y'-cos(x²)y=e* a. What are the order and degree of the differential equation? order degree b. Is the equation linear or non-linear? Explain your answer.
For the differential equation 3 y" + x*y" x² +1 Y'-cos(x²)y=e* a. What are the order and degree of the differential equation? order degree b. Is the equation linear or non-linear? Explain your answer.
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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![**For the Differential Equation**
\[ y''' + x^5 y'' + \frac{3}{x^2+1} y y' - \cos(x^2) y = e^x \]
**a.** What are the order and degree of the differential equation?
- Order: _______
- Degree: _______
**b.** Is the equation linear or non-linear? _______
Explain your answer.
---
**Explanation of Terms:**
- **Order**: The highest derivative present in the equation. In the given equation, the highest derivative is \( y''' \), so the order is 3.
- **Degree**: The power of the highest order derivative when the equation is a polynomial in derivatives. In this case, because of the term \( \frac{3}{x^2+1} y y' \) which has a product of \( y \) and \( y' \), the differential equation is not a polynomial, so the degree is not defined.
- **Linearity**: A differential equation is linear if it can be expressed as a linear combination of the derivatives of \( y \). Non-linear terms include products or powers of \( y \) and its derivatives, like \( y y' \) and \( \cos(x^2) y \), indicating the equation is non-linear.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf87c7e2-8e78-47c4-b022-e63bd8a1598e%2F75500c93-f944-4bdd-a012-b08d828f9ab3%2Fte0rqwd_processed.png&w=3840&q=75)
Transcribed Image Text:**For the Differential Equation**
\[ y''' + x^5 y'' + \frac{3}{x^2+1} y y' - \cos(x^2) y = e^x \]
**a.** What are the order and degree of the differential equation?
- Order: _______
- Degree: _______
**b.** Is the equation linear or non-linear? _______
Explain your answer.
---
**Explanation of Terms:**
- **Order**: The highest derivative present in the equation. In the given equation, the highest derivative is \( y''' \), so the order is 3.
- **Degree**: The power of the highest order derivative when the equation is a polynomial in derivatives. In this case, because of the term \( \frac{3}{x^2+1} y y' \) which has a product of \( y \) and \( y' \), the differential equation is not a polynomial, so the degree is not defined.
- **Linearity**: A differential equation is linear if it can be expressed as a linear combination of the derivatives of \( y \). Non-linear terms include products or powers of \( y \) and its derivatives, like \( y y' \) and \( \cos(x^2) y \), indicating the equation is non-linear.
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