Question

What is the difference between an ordinary differential equation and a partial differential equation?

Short answer

Answer: The main difference between ordinary differential equations (ODEs) and partial differential equations (PDEs) is that ODEs involve an unknown function and its derivatives with respect to a single independent variable, while PDEs involve an unknown function and its partial derivatives with respect to multiple independent variables. Furthermore, ODEs use regular derivatives, whereas PDEs use partial derivatives.

Step-by-step solution

Definition of Ordinary Differential Equation (ODE)

An ordinary differential equation is an equation that involves an unknown function and its derivative(s) with respect to a single independent variable. It can be represented as: F(x, y, y', y'', ..., y^{(n)}) = 0 where F(x, y, y', y'', ..., y^{(n)}) is a given function, y is the unknown function of x, and y', y'', ..., y^{(n)} are the first, second, ..., n-th derivatives with respect to x.

Definition of Partial Differential Equation (PDE)

A partial differential equation is an equation that involves an unknown function and its partial derivatives with respect to multiple independent variables. It can be represented as: G(x_1, ..., x_n, u, \frac{∂u}{∂x_1}, ..., \frac{∂u}{∂x_n}, \frac{∂^2u}{∂x_1^2}, ..., \frac{∂^2u}{∂x_n^2}, ..., \frac{∂^mu}{(∂x_1)^m}..., \frac{∂^mu}{∂x_n^m}) = 0 where G is a given function, u is the unknown function of multiple independent variables (x_1, ..., x_n), and the terms inside the parentheses represent the first, second, ..., m-th order partial derivatives with respect to the independent variables.

Comparison of ODEs and PDEs

The main differences between ordinary differential equations and partial differential equations are: 1. ODEs involve an unknown function and its derivatives with respect to a single independent variable, while PDEs involve an unknown function and its partial derivatives with respect to multiple independent variables. 2. ODEs use regular derivatives, whereas PDEs use partial derivatives. 3. Generally, ODEs' solutions are simpler and can be found through well-known methods, whereas PDEs' solutions often require more advanced mathematical techniques and are more complicated. In summary, the primary difference is that ODEs deal with functions of a single variable and their derivatives, while PDEs work with functions of multiple variables and their partial derivatives.