Give the order of the given ordinary differential equation and state the type linear or nonlinear (1 x)y" - 4xy' + 7y = cos³x "This is a [Select] equation is [Select ] and the degree of the 11
Give the order of the given ordinary differential equation and state the type linear or nonlinear (1 x)y" - 4xy' + 7y = cos³x "This is a [Select] equation is [Select ] and the degree of the 11
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
![**Transcript and Explanation for Educational Website:**
Title: Understanding Ordinary Differential Equations
Content:
**Problem Statement:**
"Give the order of the given ordinary differential equation and state the type: linear or nonlinear."
Equation:
\[
(1 - x)y'' - 4xy' + 7y = \cos^3 x
\]
Interactive Component:
- Dropdown 1: "This is a [Select] equation"
- Dropdown 2: "and the degree of the equation is [Select]"
**Explanation:**
- **Order of the Differential Equation:** The order of a differential equation is determined by the highest derivative present. In this equation, the highest derivative is \(y''\), which means the order is 2.
- **Type (Linear or Nonlinear):** A linear differential equation has terms that are linear in the dependent variable and its derivatives. In this case, the equation is linear in terms of \(y, y', \) and \(y''\). However, the presence of \(\cos^3 x\) does not affect the linearity related to \(y\).
- **Degree of the Differential Equation:** The degree is based on the power of the highest-order derivative, assuming the equation is polynomial in \(y', y'', \) etc. Here, the degree is 1 because \(y''\) is to the first power.
**Interactive Learning:**
Students are encouraged to select the correct options from the dropdowns to enhance their understanding of the classification of differential equations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1300845a-828a-484d-8f06-319293900125%2F58c69b3a-8c10-4bb3-9d2b-9cc6d6968c22%2Fl7q5vz_processed.png&w=3840&q=75)
Transcribed Image Text:**Transcript and Explanation for Educational Website:**
Title: Understanding Ordinary Differential Equations
Content:
**Problem Statement:**
"Give the order of the given ordinary differential equation and state the type: linear or nonlinear."
Equation:
\[
(1 - x)y'' - 4xy' + 7y = \cos^3 x
\]
Interactive Component:
- Dropdown 1: "This is a [Select] equation"
- Dropdown 2: "and the degree of the equation is [Select]"
**Explanation:**
- **Order of the Differential Equation:** The order of a differential equation is determined by the highest derivative present. In this equation, the highest derivative is \(y''\), which means the order is 2.
- **Type (Linear or Nonlinear):** A linear differential equation has terms that are linear in the dependent variable and its derivatives. In this case, the equation is linear in terms of \(y, y', \) and \(y''\). However, the presence of \(\cos^3 x\) does not affect the linearity related to \(y\).
- **Degree of the Differential Equation:** The degree is based on the power of the highest-order derivative, assuming the equation is polynomial in \(y', y'', \) etc. Here, the degree is 1 because \(y''\) is to the first power.
**Interactive Learning:**
Students are encouraged to select the correct options from the dropdowns to enhance their understanding of the classification of differential equations.
Expert Solution
Step 1
The order of a differential equation is the highest order of the derivative it contains.
Degree of a differential equation is the power of the highest order derivative.
Step by step
Solved in 3 steps with 2 images
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