Example of First-Order Differential Equation: Mathematical Instance

What is an example of a first order differential equation?

A first-order differential equation involves the first derivative of an unknown function with respect to the independent variable. The general form of a first-order linear ordinary differential equation (ODE) is often represented as:

$\frac{dy}{dx} = f(x, y)$

Here, $y$ is the dependent variable, $x$ is the independent variable, and $f(x, y)$ is a given function.

Let's consider a specific example of a first-order linear differential equation:

$\frac{dy}{dx} = 2x$

In this case, the rate of change of the function $y$ with respect to $x$ is given by $2x$. To find the solution to this differential equation, you would integrate both sides with respect to $x$:

$\int \frac{dy}{dx} \, dx = \int 2x \, dx$

The solution to this equation, after performing the integrations, would be:

$y(x) = x^2 + C$

Here, $C$ is the constant of integration, which would be determined by any initial conditions provided or through additional information about the problem.

So, the first-order differential equation $\frac{dy}{dx} = 2x$ has the solution $y(x) = x^2 + C$.

dP/dt = kP

dy/dx = f(x, y)