Question

Classify each of the following differential equations as ordinary or partial differential equations; state the order of each equation; and determine whether the equation under consideration is linear or nonlinear. $$ \frac{\partial^{4} u}{\partial x^{2} \partial y^{2}}+\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}+u=0 $$

Short answer

The given differential equation is a linear partial differential equation (PDE) of order 4, as it contains partial derivatives and has no products or powers of the dependent variable \(u\) and its derivatives.

Step-by-step solution

Identify the type of differential equation

First, we can identify if the given equation is an ODE or PDE by checking the types of derivatives present. If there are partial derivatives, then the equation is a PDE; otherwise, it is an ODE. In this case, we have partial derivatives with respect to \(x\) and \(y\), so the given equation is a partial differential equation (PDE).

Determine the order of the differential equation

To determine the order, we can find the highest derivative in the equation. In our given PDE, we have \(\frac{\partial^{4} u}{\partial x^{2} \partial y^{2}}\), which is a fourth-order partial derivative. Therefore, the order of our PDE is 4.

Check if the differential equation is linear or nonlinear

A differential equation is considered linear if there are no products or powers of the dependent variable and its derivatives. In our case, the dependent variable is \(u\), and the given differential equation is: \[ \frac{\partial^{4} u}{\partial x^{2} \partial y^{2}}+\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}+u=0 \] There are no products or powers of \(u\) and its derivatives, so the PDE is linear. #Summary#: The given differential equation is a linear partial differential equation of order 4.

Key concepts

Partial Differential Equations

In the world of differential equations, distinguishing between various types is crucial. A partial differential equation (PDE) involves functions of more than one variable and their partial derivatives. This is different from ordinary differential equations (ODEs), which involve functions of a single variable and their ordinary derivatives.

To recognize a PDE, look for partial derivatives in the expression. Partial derivatives are denoted by the symbol \( \partial \), indicating that the function is differentiated with respect to one variable while keeping others constant. For example, in the given PDE:

• \( \frac{\partial^{4} u}{\partial x^{2} \partial y^{2}} \)
• \( \frac{\partial^{2} u}{\partial x^{2}} \)
• \( \frac{\partial^{2} u}{\partial y^{2}} \)

Each term shows partial derivatives of the function \( u \), indicating the involvement of multiple variables, \( x \) and \( y \). Therefore, the equation is classified as a PDE. Understanding this classification helps in deciding which mathematical techniques to apply for solving the equation.

Linear Equations

The linearity of a differential equation is another significant classification criterion. A linear differential equation ensures a certain simplicity and predictability in behavior, making it easier to solve analytical or numerical methods.

For a differential equation to be linear, it must meet specific requirements:

• The dependent variable and its derivatives appear only to the first power.
• No products of the dependent variable and its derivatives occur.
• The coefficients of the variables and derivatives can be functions of the independent variables, but not of the dependent variable.

Applying these criteria to our case, the differential equation:\[\frac{\partial^{4} u}{\partial x^{2} \partial y^{2}} + \frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} + u = 0\]This equation meets the criteria, confirming it's linear. Linear equations are pivotal in understanding more complex nonlinear models and are used widely in engineering and physics.

Order of Differential Equations

The order of a differential equation is determined by the highest derivative that appears in the equation. This concept is essential as it often dictates the complexity and method to solve the equation.

In the context of the given PDE, we examine the derivatives:

• One term, \( \frac{\partial^{4} u}{\partial x^{2} \partial y^{2}} \), specifies a fourth-order derivative.
• Other terms have second-order derivatives: \( \frac{\partial^{2} u}{\partial x^{2}} \) and \( \frac{\partial^{2} u}{\partial y^{2}} \).

Since the highest derivative present is fourth-order, this determines the equation's order as 4. Knowing the order is crucial because it not only gives insight into the equation's characteristics but also determines which boundary conditions and methods to apply when solving the PDE.