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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Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
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![The equation shown is a first-order linear differential equation:
\[ xy' + (1 + x)y = e^x \]
### Explanation:
- **\(xy'\)**: This term indicates the derivative of \(y\) with respect to \(x\), multiplied by \(x\).
- **\((1 + x)y\)**: This term represents \(y\) multiplied by the expression \(1 + x\).
- **\(e^x\)**: This is the exponential function, which serves as the non-homogeneous part of the equation.
In differential equations, solving a linear equation typically involves finding an integrating factor, which can be used to simplify and solve the equation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffe11b9c2-33d9-41e3-b9ce-693255816c85%2F27be8de4-9265-4eec-a8fd-871c062867ed%2Fb7pqe3w_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The equation shown is a first-order linear differential equation:
\[ xy' + (1 + x)y = e^x \]
### Explanation:
- **\(xy'\)**: This term indicates the derivative of \(y\) with respect to \(x\), multiplied by \(x\).
- **\((1 + x)y\)**: This term represents \(y\) multiplied by the expression \(1 + x\).
- **\(e^x\)**: This is the exponential function, which serves as the non-homogeneous part of the equation.
In differential equations, solving a linear equation typically involves finding an integrating factor, which can be used to simplify and solve the equation.
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