The partial differential equation $\frac{{\partial }^2\varphi }{\partial x^2}+\frac{{\partial }^2\varphi }{\partial y^2}+\left(\frac{\partial \varphi }{\partial x}\right)+\left(\frac{\partial \varphi }{\partial y}\right)=0$ has

Understanding the characteristics of a partial differential equation (PDE) involves identifying its order and degree. These two properties are fundamental to classifying and solving PDEs. Let's analyze the given partial differential equation:

$$\frac{{\partial }^2\varphi }{\partial x^2}+\frac{{\partial }^2\varphi }{\partial y^2}+\left(\frac{\partial \varphi }{\partial x}\right)+\left(\frac{\partial \varphi }{\partial y}\right)=0$$

Partial Differential Equation Definition

A partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. PDEs are essential mathematical tools used to model various phenomena in physics, engineering, economics, and other scientific fields.

To accurately describe a partial differential equation, two key properties are considered: its order and its degree. Let's define these terms clearly:

- Order of a Partial Differential Equation: The order of a partial differential equation is determined by the highest order of any partial derivative present in the equation. For example, if the highest derivative is a second derivative, the order is 2.
- Degree of a Partial Differential Equation: The degree of a partial differential equation is the power of the highest order partial derivative, provided that the equation is a polynomial in its derivatives. It is crucial to ensure that the equation is free from radicals or fractions involving the derivatives before determining the degree. If the equation is not a polynomial in its derivatives, its degree is undefined.

Order of the Given Partial Differential Equation

Let's carefully examine each term in the given partial differential equation to identify the orders of the partial derivatives involved:

- The term $\frac{{\partial }^2\varphi }{\partial x^2}$ involves a second-order partial derivative of $\varphi$ with respect to $x$.
- The term $\frac{{\partial }^2\varphi }{\partial y^2}$ involves a second-order partial derivative of $\varphi$ with respect to $y$.
- The term $\frac{\partial \varphi }{\partial x}$ involves a first-order partial derivative of $\varphi$ with respect to $x$.
- The term $\frac{\partial \varphi }{\partial y}$ involves a first-order partial derivative of $\varphi$ with respect to $y$.

By comparing these terms, we can see that the highest order of partial derivatives present in the partial differential equation is 2. Therefore, the order of the given partial differential equation is 2.

Degree of the Given Partial Differential Equation

Now, let's determine the degree of the partial differential equation. The degree is the power of the highest order derivative term. In our equation, the highest order derivatives are the second-order derivatives: $\frac{{\partial }^2\varphi }{\partial x^2}$ and $\frac{{\partial }^2\varphi }{\partial y^2}$.

Let's observe the powers of these highest order terms:

- The term $\frac{{\partial }^2\varphi }{\partial x^2}$ is raised to the power of 1.
- The term $\frac{{\partial }^2\varphi }{\partial y^2}$ is also raised to the power of 1.

The entire partial differential equation is already in a form where all derivatives are free from fractional or radical powers, meaning it is a polynomial in its derivatives. The highest order terms (second-order derivatives) are both raised to the power of 1. Therefore, the degree of the given partial differential equation is 1.

Summary of Partial Differential Equation Properties

To summarize our analysis of the partial differential equation:

- The Order of the partial differential equation is 2.
- The Degree of the partial differential equation is 1.

This matches the description "Degree 1 order 2".