# 4: First Order Ordinary Differential Equations

• • Contributed by Marcia Levitus

Objectives

• Be able to identify the dependent and independent variables in a differential equation.
• Be able to identify whether an ordinary differential equation (ODE) is linear or nonlinear.
• Be able to identify the order of an ODE
• Be able to identify whether a first order ODE is separable or not.
• Be able to find the general and particular solutions of linear first order ODEs.
• Be able to find the general and particular solutions of separable first order ODEs.
• Understand how to verify that the solution you got in a problem satisfies the differential equation and initial conditions.
• Understand how to solve differential equations in the context of chemical kinetics. Understand the concept of mass balance, and half-life.

• 4.1: Definitions and General Concepts
A differential equation is an equation that defines a relationship between a function and one or more derivatives of that function.  An ordinary differential equation (ODE) relates an unknown function, y(t) as a function of a single variable. Differential equations arise in the mathematical models that describe most physical processes.
• 4.2: 1st Order Ordinary Differential Equations
We will discuss only two types of 1st order ODEs, which are the most common in the chemical sciences: linear 1st order ODEs, and separable 1st order ODEs. These two categories are not mutually exclusive, meaning that some equations can be both linear and separable, or neither linear nor separable.
• 4.3: Chemical Kinetics
The term chemical kinetics refers to the study of the rates of chemical reactions. Differential equations play a central role in the mathematical treatment of chemical kinetics. We will start with the simplest examples, and then we will move to more complex cases. We will focus on a couple of reaction mechanisms. The common theme will be to find expressions that will allow us to calculate the concentration of the different species that take part of the reaction at different reaction times.
• 4.4: Problems