The differential equation has an implicit general solution of the form \( F(x, y)=K \), where \( K \) is an arbitary constant. In fact, because the differential equation is separable, we can define the solution curve implicitly by a function in the form \[ F(x, y)=G(x)+H(y)=K . \] Find such a solution and then give the related functions requested. \( F(x, y)=G(x)+H(y)= \)

To find the implicit general solution of a separable differential equation in the form \( F(x, y) = G(x) + H(y) = K \), we need to follow these steps: 1. **Identify the differential equation**: Since the specific differential equation is not provided, I will assume a general separable form \( \frac{dy}{dx} = f(x)g(y) \). 2. **Separate the variables**: Rearranging gives us \( \frac{1}{g(y)} dy = f(x) dx \). 3. **Integrate both sides**: We will integrate the left side with respect to \( y \) and the right side with respect to \( x \). 4. **Define the functions \( G(x) \) and \( H(y) \)**: After integration, we will express the solution in the form \( F(x, y) = G(x) + H(y) = K \). Since the specific form of the differential equation is not given, I will proceed with a general example. Let's assume the differential equation is: \[ \frac{dy}{dx} = xy \] ### Step 1: Separate the variables \[ \frac{1}{y} dy = x dx \] ### Step 2: Integrate both sides Integrating both sides gives: \[ \int \frac{1}{y} dy = \int x dx \] This results in: \[ \ln |y| = \frac{x^2}{2} + C \] ### Step 3: Solve for \( y \) Exponentiating both sides, we have: \[ |y| = e^{\frac{x^2}{2} + C} = e^C e^{\frac{x^2}{2}} \] Let \( K = e^C \), then: \[ y = K e^{\frac{x^2}{2}} \] ### Step 4: Define \( G(x) \) and \( H(y) \) We can express this in the form \( F(x, y) = G(x) + H(y) = K \): - Let \( G(x) = \frac{x^2}{2} \) - Let \( H(y) = \ln |y| \) Thus, the implicit solution can be written as: \[ F(x, y) = \frac{x^2}{2} + \ln |y| = K \] If you have a specific differential equation in mind, please provide it, and I can give a more tailored solution.