What is a separable first-order differential equation?

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1. A separable first-order differential equation is a type of differential equation that can be written in the form \(\frac{dy}{dx} = g(x)h(y)\), where the right-hand side is a product of a function of \(x\) and a function of \(y\).
2. The key idea is that the variables \(x\) and \(y\) can be separated on opposite sides of the equation, allowing us to rewrite it as \(\frac{1}{h(y)} dy = g(x) dx\).
3. Once separated, we integrate both sides with respect to their own variables: \(\int \frac{1}{h(y)} dy = \int g(x) dx\).
4. After performing the integrations, we obtain an implicit or explicit solution involving \(x\) and \(y\).
5. This method is useful because it transforms the differential equation into two simpler integrals, making it easier to solve.

Key Concepts

First-Order Differential Equation

A first-order differential equation involves the first derivative of an unknown function with respect to an independent variable. It expresses a relationship between the function and its rate of change, typically written as dy/dx = f(x, y).

Separable Differential Equation

A separable differential equation is one where the variables can be separated on opposite sides of the equation, allowing it to be written as g(y) dy = h(x) dx. This form enables integration of each side independently to find the solution.

Method of Separation of Variables

The method of separation of variables solves separable equations by isolating y terms with dy and x terms with dx, then integrating both sides. This technique transforms a differential equation into two integrals, simplifying the process of finding explicit solutions.