I believe that differential equations are an important part of mathematics and science. While the concept is broad, there are two main types of differential equations that you need to know about ordinary differential equations and partial differential equations.
So, what is the difference between ordinary differential equations and partial differential equations? Ordinary differential equations (ODE) are equations where we take the derivatives with respect to only one variable with only one independent variable. In contrast, partial differential equations (PDE) typically depend on partial derivatives of several variables (Source: University of Victoria)
Read on to find out the differences between ordinary differential equations (ODEs) and partial differential equations (PDEs).
What Are Ordinary Differential Equations (ODEs)?
An ordinary differential equation (ODE) is a type of differential equation that involves a single independent variable and its derivatives (e.g., x, dx/dt). It describes how a function changes as time passes or as other variable changes, such as temperature or pressure. These equations are used to model physical phenomena such as gravity, electric current, the motion of objects, velocity, and acceleration.
ODEs can be solved analytically or numerically, depending on their complexity. Furthermore, ODEs are relatively simple to solve since they involve only one derivative. Generally speaking, these equations can be solved using numerical methods such as Runge-Kutta or Euler’s method.
I encourage you to watch the video below to learn Ordinary Differential Equations.
What Are Partial Differential Equations (PDEs)?
A partial differential equation (PDE) is a type of differential equation that involves multiple independent variables and their derivatives (e.g., x, y, z). PDEs are used to model phenomena such as heat transfer and fluid flow over multiple spatial dimensions.
Generally, PDEs help describe how systems evolve over time and space; for example, heat transfer within an object or population growth over a large area. These equations can be difficult to solve compared to ODEs since they require multiple derivatives instead of just one.
If you are interested in learning more about partial differential equations I encourage you to watch the video below.
Differences Between Ordinary Differential Equations (ODEs) And Partial Differential Equations (PDEs)
Unlike, PDEs involve more than one independent variable and its derivatives. This means that the equation has two or more variables in it that can change simultaneously with time or some other factor; for example, temperature over space and time.
As with ODEs, PDEs can also be solved analytically or numerically, depending on their complexity. In general, PDEs are much more difficult to solve than ODEs because they involve multiple variables and derivatives instead of just one variable and derivative, like in the case of ODEs.
Additionally, Partial Differential Equations (PDEs) are typically used to model functions with many factors precisely.
Suppose we want to deal with temperature varying depending on time. In other words, how likely is the temperature to change with respect to time; we take derivative w. r. t time by keeping other parameters as constants (lat, lag, etc.).
The main difference is that ODEs involve a single independent variable, while PDEs involve more than one independent variable, making PDEs much more complex than ODEs. Furthermore, PDEs require multiple derivatives rather than just one, as ODEs do.
Differential equations play an important role in mathematics and science as they allow us to model physical phenomena in ways that would otherwise be impossible or impractical to do so with traditional mathematical tools alone.
The two main types of differential equations include ordinary differential equations (ODEs) and partial differential equations (PDEs), and I believe that each has its own set of advantages and disadvantages when it comes to solving them analytically or numerically.
I believe that understanding the differences between these two types will help you choose the right approach for your specific problem.