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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![**Solving Differential Equations: An Example**
In this section, we will solve a given differential equation through integration. Consider the following differential equation:
\[
\frac{dy}{y} = \frac{-dx}{x}
\]
### Step-by-Step Solution
1. **Separate the Variables**:
- To solve the differential equation, the first step is to separate the variables \( y \) and \( x \) on opposite sides of the equation.
\[
\frac{dy}{y} = \frac{-dx}{x}
\]
2. **Integrate Both Sides**:
- Integrate the left side of the equation with respect to \( y \) and the right side with respect to \( x \):
\[
\int \frac{1}{y} \, dy = \int \frac{-1}{x} \, dx
\]
3. **Evaluate the Integrals**:
- The integral of \(\frac{1}{y}\) with respect to \( y \) is \(\ln |y|\). The integral of \(\frac{-1}{x}\) with respect to \( x \) is \(-\ln |x|\):
\[
\ln |y| = -\ln |x| + C
\]
Here, \( C \) is the constant of integration.
4. **Simplify the Equation**:
- You can combine the logarithms and solve for \( y \):
\[
\ln |y| + \ln |x| = C
\]
\[
\ln |xy| = C
\]
Exponentiating both sides to remove the logarithm, we get:
\[
|xy| = e^C
\]
Since \( e^C \) is simply a constant, it can be replaced with any positive constant \( K \):
\[
|xy| = K
\]
5. **Final Solution**:
- Therefore, the general solution to the differential equation is:
\[
xy = \pm K
\]
or equivalently,
\[
xy = C
\]
where \( C \) is a constant.
The solved equation represents a family of hyperbolas in the \(xy\)-plane. This example demonstrates the method of](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9973d75d-32e3-4ebe-9432-aeaa4634e542%2F7614a46a-66f4-4183-a95d-b7ab9f7a32d8%2Fnxqu9ll_processed.png&w=3840&q=75)
Transcribed Image Text:**Solving Differential Equations: An Example**
In this section, we will solve a given differential equation through integration. Consider the following differential equation:
\[
\frac{dy}{y} = \frac{-dx}{x}
\]
### Step-by-Step Solution
1. **Separate the Variables**:
- To solve the differential equation, the first step is to separate the variables \( y \) and \( x \) on opposite sides of the equation.
\[
\frac{dy}{y} = \frac{-dx}{x}
\]
2. **Integrate Both Sides**:
- Integrate the left side of the equation with respect to \( y \) and the right side with respect to \( x \):
\[
\int \frac{1}{y} \, dy = \int \frac{-1}{x} \, dx
\]
3. **Evaluate the Integrals**:
- The integral of \(\frac{1}{y}\) with respect to \( y \) is \(\ln |y|\). The integral of \(\frac{-1}{x}\) with respect to \( x \) is \(-\ln |x|\):
\[
\ln |y| = -\ln |x| + C
\]
Here, \( C \) is the constant of integration.
4. **Simplify the Equation**:
- You can combine the logarithms and solve for \( y \):
\[
\ln |y| + \ln |x| = C
\]
\[
\ln |xy| = C
\]
Exponentiating both sides to remove the logarithm, we get:
\[
|xy| = e^C
\]
Since \( e^C \) is simply a constant, it can be replaced with any positive constant \( K \):
\[
|xy| = K
\]
5. **Final Solution**:
- Therefore, the general solution to the differential equation is:
\[
xy = \pm K
\]
or equivalently,
\[
xy = C
\]
where \( C \) is a constant.
The solved equation represents a family of hyperbolas in the \(xy\)-plane. This example demonstrates the method of
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