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The image contains a first order differential equation of the form \( y' = \frac{f(x)}{g(y)} \). This type of equation is separable, which means we can rewrite it in a form that allows us to separate the variables \( x \) and \( y \) on different sides of the equation. Here's how you can solve it step-by-step:

### Step-by-Step Solution

1. **Separate Variables:** Start by rewriting the equation to separate \( x \) and \( y \):
   \[
   g(y) \, dy = f(x) \, dx
   \]

2. **Integrate Both Sides:** Integrate both sides with respect to their respective variables:
   \[
   \int g(y) \, dy = \int f(x) \, dx
   \]
   This step will yield two antiderivatives, one in terms of \( y \) and the other in terms of \( x \).

3. **Solve for \( y \):** The integration may involve constants of integration. Assuming constants are merged or one is subtracted from the other, we can represent it as \( C \). If possible, solve the resulting equation for \( y \) in terms of \( x \).

4. **Initial Condition (if given):** If an initial condition is provided, you can use it to solve for the constant \( C \).

### Example

If \( f(x) = x \) and \( g(y) = y \), the separable form becomes:
\[
y \, dy = x \, dx
\]
Integrating both sides:
\[
\int y \, dy = \int x \, dx \implies \frac{y^2}{2} = \frac{x^2}{2} + C
\]
Solving for \( y \):
\[
y = \pm \sqrt{x^2 + 2C}
\]

Would you like to go through a specific example with actual functions for \( f(x) \) and \( g(y) \)? Or do you have any further questions about this process?

#### Additional Questions
1. What are the applications of separable differential equations?
2. How can you verify the solution of a differential equation?
3. What happens if \( g(y) \) or \( f(x) \) includes a term that makes the integral difficult or impossible to solve analytically?
4. Can separable differential equations model exponential growth or decay?
5. Are there any special techniques needed if \( g(y) \) is a function that leads to a singular solution at certain values of \( y \)?

#### Tip
When solving differential equations, always check if the variables can be separated as it simplifies the solving process.

A first order differential equation of the form y' = f(x) / g(y).

Learn how to solve first order differential equations of the form y' = f(x) / g(y) using the separation of variables technique. This page provides a step-by-step guide with examples, covering key mathematical concepts such as integration and differential equations. Suitable for undergraduate mathematics students.

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