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.