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. $$ \frac{d^{6} x}{d t^{6}}+\left(\frac{d^{4} x}{d t^{4}}\right)\left(\frac{d^{3} x}{d t^{3}}\right)+x=t $$

Short answer

The given differential equation is an ordinary differential equation of order 6 and is nonlinear.

Step-by-step solution

Identify if the differential equation is ordinary or partial.

An ordinary differential equation contains only one independent variable and its derivatives. A partial differential equation contains more than one independent variable and their partial derivatives. In our case, we can see that the given equation contains only one independent variable (t) and its derivatives. Therefore, the given differential equation is an ordinary differential equation.

Find the order of the differential equation.

The order of a differential equation is the highest power of the derivative present in the equation. In the given equation, the highest power of the derivative is 6: \(\frac{d^{6} x}{d t^{6}}\) So, the order of the given differential equation is 6.

Determine the linearity of the differential equation.

A linear differential equation is one in which the dependent variable (x) and its derivatives appear linearly (raised to the power of 1). If the dependent variable or any of its derivatives is raised to a power other than 1, or multiplied by another derivative, the equation is nonlinear. The given equation is: \( \frac{d^{6} x}{d t^{6}}+\left(\frac{d^{4} x}{d t^{4}}\right)\left(\frac{d^{3} x}{d t^{3}}\right)+x=t \) It can be observed that the second term involves the multiplication of two derivatives: \(\left(\frac{d^{4} x}{d t^{4}}\right)\left(\frac{d^{3} x}{d t^{3}}\right)\) This term makes the equation nonlinear as it is not in the linear form. Therefore, the given differential equation is nonlinear. To summarize our results: 1. The given differential equation is an ordinary differential equation. 2. The order of the equation is 6. 3. The given differential equation is nonlinear.

Key concepts

Differential Equation Classification

A differential equation can be an ordinary differential equation (ODE) or a partial differential equation (PDE). Understanding the distinction between these two types is crucial in solving them correctly.

• An **ordinary differential equation (ODE)** involves derivatives with respect to a single independent variable, such as time \(t\). This is the case for our example equation, which only includes derivatives of \(x\) with respect to \(t\).
• **Partial differential equations (PDEs)**, on the other hand, have derivatives involving more than one independent variable. Each independent variable might represent dimensions like time, space, or other parameters.

Recognizing whether a differential equation is ordinary or partial helps determine which mathematical techniques and tools are appropriate for finding solutions.

Differential Equation Order

The order of a differential equation is a straightforward concept. It tells us the highest derivative involved in the equation. Finding the order helps identify the complexity of the equation.

• The **order** is determined by finding the derivative of the highest degree. In our example, \( \frac{d^{6} x}{d t^{6}} \) is the term with the highest derivative, making the order of the equation six.

Knowing the order is important because it influences how we approach finding solutions. Higher-order equations can be more complex and may require more advanced techniques or more information (like auxiliary conditions) to solve.

Linearity in Differential Equations

In the context of differential equations, understanding linearity is essential as it dictates how solutions are derived.

• A **linear differential equation** means that the dependent variable and all its derivatives appear to the first power and are not multiplied together. This makes them more straightforward to solve because they follow linear algebra rules.
• A **nonlinear differential equation** includes terms where the dependent variable or its derivatives are raised to higher powers or are products of each other. Our example equation is nonlinear since it has a term \( \left(\frac{d^{4} x}{d t^{4}}\right)\left(\frac{d^{3} x}{d t^{3}}\right) \), involving a product of two derivatives.

Linear equations tend to have well-behaved solutions, while nonlinear equations may show complex phenomena like chaos. Understanding the linearity of an equation guides us in selecting the appropriate mathematical tools to apply.