First-Order Ordinary Differential Equations

First-Order Ordinary Differential Equations

Ordinary differential equations, sometimes abbreviated as ODEs, are equations comprised of a function containing an unknown and the derivatives of that function. Since the order of an ordinary differential equation is the order of the highest-order derivative of the function appearing in the equation a first-order ordinary differential equation is one that contains the first derivative of the function. They commonly have the form: dy/dx = f(x, y), where f(x, y) is a function of x and y, and dy/dx is the first derivative of that function with respect to x. A solution to a first-order ordinary differential equation is any function y that satisfies that differential equation. First-order ordinary differential equations have one linearly independent solution. They can describe the change in the size of a population, the motion of a falling body, the flow of current in an electric circuit, and the motion of a pendulum just to mention a few. First-order ordinary differential equations have a wide variety of uses and as such are used in a variety of fields such as chemistry, physics, economics, engineering, and electronics.

There are different types of first-order ordinary differential equations described as linear, exact, separable, homogeneous or cross multiple. The methods of finding solutions depend upon the classification of the specific equation. If a first-order ordinary differential equation has the form: dy/dx + P(x)y = Q(x) where P and Q are continuous then it is said to be a linear first-order differential equation. This type of equation can be solved by the method of finding an integrating factor. This method involves finding a function by which the original ordinary differential equation can be multiplied to make it integrable. When Q(x) = 0 the linear first-order differential equation is said to be homogeneous as well. If the linear first-order ordinary differential equation is nonhomogeneous, if Q(x) 0, then another method called variation of parameters can be employed to find the solution.

If a first-order differential equation is of the form: p(x, y)dx + q(x, y)dy = 0 and ∂p/∂y = ∂q/∂x then it is said to be exact and means that a conservative field exists and that a scalar potential can be defined. The solution to a first-order ordinary differential equation that is exact is given by: If ∂p/∂y ∂q/∂x then the first-order ordinary differential equation is said to be inexact and may be solved by defining an integrating factor so that the resulting equation becomes exact and can be solved as above.

If a first-order ordinary differential equation can be expressed as: dy/dx = X(x)Y(y) it is called a separable equation and can be solved using a technique called separation of variables. This involves rearranging the above form and integrating both sides to yield the solution: dy/Y(y) = X(x)dx.

Although there are many methods for solving the different classes of first-order ordinary differential equations the only practical solution method for very complicated equations is to use numerical methods. All solutions to first-order ordinary differential equations satisfy existence and uniqueness properties. That is if one solution can be found for a problem then it is said that a solution exists and that it is the only solution. The methods for solving first-order ordinary differential equations are very important because every higher order ordinary differential equation can be expressed as a system of first-order differential equations. In some cases it is helpful to break a higher order differential equation down into a system of first-order equations and to solve them.