Solve the differential equation: $$\frac{dy}{dx} = \frac{x}{y}$$

$$y = \text{±}\text{sqrt}(x^2 + C)$$ .

Provide an example of a separable differential equation.

$$\frac{dy}{dx} = y(1 - y)$$ is separable because it can be written as $$\frac{1}{y(1 - y)}dy = dx$$ .

What does the constant $$C$$ represent in the solution of a differential equation?

The constant $$C$$ represents the arbitrary constant of integration, which accounts for the family of solutions.

How do you verify if a differential equation is separable?

Check if it can be expressed as $$\frac{dy}{dx} = g(y)h(x)$$ , allowing separation of variables.

What is the significance of the initial condition in solving differential equations?

The initial condition allows us to determine the specific value of the constant $$C$$ in the general solution.

What is the method of integrating factors?

The method of integrating factors is used for solving linear differential equations, not separable ones.

What is the role of integration in solving separable differential equations?

Integration is used to find the general solution after separating the variables.

What is the difference between a homogeneous and a separable differential equation?

A homogeneous differential equation can be expressed as $$\frac{dy}{dx} = figg(\frac{y}{x}igg)$$ , while a separable equation can be expressed as $$\frac{dy}{dx} = g(y)h(x)$$ .

Differential Equations - Separable Variable Method Flashcards

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What is a separable differential equation?

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Separable.

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