dy dx -y dx + (x+ /xy) dy =e (x+/xy
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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solve the given
appropriate substitution
![**Topic: Differential Equations**
**Example Problem: Solving a First Order Differential Equation**
Consider the following differential equation:
\[ \frac{dy}{dx} = \frac{y - x}{y + x} \]
To solve this, we perform a variable separation and rearrange the equation. An implicit form of this equation can be written as follows:
\[ -y \, dx + \left( x + \sqrt{xy} \right) \, dy = 0 \]
Here, a differential equation is presented in the form of a first order, and the given form separates the differentials \( dx \) and \( dy \), allowing us to solve it using appropriate techniques such as integration. Let's break down the steps to solve this type of differential equation.
1. **Identify and Separate Variables:**
- Start by isolating the differentials \( dx \) and \( dy \).
- Rewrite the equation in a format that allows integration with respect to each variable.
2. **Integrate Both Sides:**
- Integrate each side of the equation with respect to its variable.
3. **Solve for the Dependent Variable:**
- Solve the resulting equation for the dependent variable \( y \) in terms of \( x \).
4. **Determine the General Solution:**
- Include the constant of integration and specify any initial conditions if provided, which will help find a particular solution.
This process will help you solve first order differential equations encountered in calculus and related fields.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1e4fcbe0-ed74-4447-b38a-a2af13f9fabc%2F0a79301d-aab5-44b7-8359-02e4437b86f9%2Ftxp564l_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Topic: Differential Equations**
**Example Problem: Solving a First Order Differential Equation**
Consider the following differential equation:
\[ \frac{dy}{dx} = \frac{y - x}{y + x} \]
To solve this, we perform a variable separation and rearrange the equation. An implicit form of this equation can be written as follows:
\[ -y \, dx + \left( x + \sqrt{xy} \right) \, dy = 0 \]
Here, a differential equation is presented in the form of a first order, and the given form separates the differentials \( dx \) and \( dy \), allowing us to solve it using appropriate techniques such as integration. Let's break down the steps to solve this type of differential equation.
1. **Identify and Separate Variables:**
- Start by isolating the differentials \( dx \) and \( dy \).
- Rewrite the equation in a format that allows integration with respect to each variable.
2. **Integrate Both Sides:**
- Integrate each side of the equation with respect to its variable.
3. **Solve for the Dependent Variable:**
- Solve the resulting equation for the dependent variable \( y \) in terms of \( x \).
4. **Determine the General Solution:**
- Include the constant of integration and specify any initial conditions if provided, which will help find a particular solution.
This process will help you solve first order differential equations encountered in calculus and related fields.
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