First-order differential equations play a crucial role in calculus as they describe the relationship between a function, its derivative, and an independent variable. These equations are called "first-order" because they involve only the first derivative of the unknown function. They have widespread applications in mathematics, physics, engineering, biology, economics, and more.
A first-order ordinary differential equation (ODE) is expressed as:
dy/dx = f(x, y)
where dy/dx represents the derivative of the unknown function y with respect to the independent variable x, and f(x, y) is a given function that relates the derivative of y to x and y.
One common technique for solving first-order differential equations is separation of variables. This method involves rearranging the equation so that all terms containing y are on one side and all terms containing x are on the other side. The separated equation can then be integrated to find the general solution.
Consider the following first-order differential equation:
dy/dx = x^2 + y^2
To solve this equation using separation of variables, we first separate the variables by moving terms containing y to one side:
dy / (y^2) = (x^2) dx
Next, we integrate both sides of the equation:
∫(1/y^2) dy = ∫x^2 dx
Integrating the left side:
-1/y = (1/3) x^3 + C
where C is the constant of integration. Solving for y, we get:
y = -1 / ((1/3) x^3 + C)
This is the general solution to the given first-order differential equation.
In this example, we separated the variables, integrated each side, and solved for y to obtain the general solution. The constant of integration C arises due to the indefinite integral.
First-order differential equations and the separation of variables technique provide a powerful toolset for solving various mathematical problems. In the next post, we will delve deeper into the concept of separable differential equations and understand their significance.