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. $$ \frac{d^{3} y}{d x^{3}}+4 \frac{d^{2} y}{d x^{2}}-5 \frac{d y}{d x}+3 y=\sin x $$

Short answer

The given differential equation is a third-order linear ordinary differential equation.

Step-by-step solution

Identify Derivatives

In the given equation, \[ \frac{d^{3} y}{d x^{3}}+4 \frac{d^{2} y}{d x^{2}}-5 \frac{d y}{d x}+3 y=\sin x \] we see that it contains ordinary derivatives of \(y\) with respect to \(x\).

Classify as Ordinary or Partial Differential Equation

Since the given equation contains ordinary derivatives, it can be classified as an ordinary differential equation (ODE).

Determine the Order

To find the order of the differential equation, we look at the highest order of the ordinary derivatives present in the equation. In this case, we have: \[ \frac{d^{3} y}{d x^{3}} \] which is a third-order derivative. So, the given equation is a third-order ODE.

Determine if Linear or Nonlinear

To determine if the equation is linear or nonlinear, we will check if it follows the standard form of a linear equation, which for a third-order ODE is: \[ a_{3}\frac{d^{3} y}{d x^{3}}+a_{2}\frac{d^{2} y}{d x^{2}}+a_{1}\frac{d y}{d x}+a_{0}y=g(x) \] Comparing this with our given equation, we see that it does follow the above form, with \(a_{3} = 1\), \(a_{2} = 4\), \(a_{1} = -5\), \(a_{0} = 3\), and \(g(x) = \sin x\). Hence, the equation is linear. In conclusion, the given differential equation is a third-order linear ordinary differential equation.

Key concepts

Linear Differential Equations

A linear differential equation is characterized by the fact that the dependent variable and all its derivatives appear to the first power, and they are not multiplied together. This means there are no products of the function or its derivatives, like \(y^2\) or \(y \, \frac{d y}{d x}\), within the equation.

The general form of a linear differential equation is often presented as:

• \(a_{n}(x)\frac{d^{n} y}{d x^{n}} + a_{n-1}(x)\frac{d^{n-1} y}{d x^{n-1}} + \ldots + a_{1}(x)\frac{d y}{d x} + a_{0}(x)y = g(x)\)

Where \(a_{0}, a_{1}, \ldots, a_{n}\) are coefficients that can be functions of \(x\), and \(g(x)\) is a function usually representing input or forcing terms.

In the context of the given exercise, the equation follows this standard form:

• \(1 \frac{d^{3} y}{d x^{3}} + 4 \frac{d^{2} y}{d x^{2}} - 5 \frac{d y}{d x} + 3 y = \sin x\)

Here, the coefficients are constants, and the output \(g(x)\) is the function \(sin x\).Additionally, the characteristics make it easy to identify, among the broader class of differential equations, which are linear.

ODE Classification

Ordinary Differential Equations (ODEs) involve functions of a single independent variable and their derivatives. They mainly contrast with partial differential equations, which involve multiple independent variables.

For an equation to be classified as an ODE, all the derivatives present have to be with respect to one variable. In our case, the expression involves \(y\) as a function of \(x\), making it an ODE.

Here's a quick way to identify ODEs:

• Check if all derivatives are ordinary, meaning each derivative includes differentiation with respect to a single variable.
• If the equation does not involve differences in more than one variable, it's not a partial differential equation (PDE).

In this exercise, since there are no terms involving partial derivatives (differentiate with respect to various variables), we classify the equation as an ODE. This classification helps in understanding the approach to solve the equation efficiently.

Third-Order Derivatives

Derivatives are a fundamental concept in calculus that represent the rate of change at a particular point. A third-order derivative provides the rate of rate of rate of change of the function, which is quite a detailed look into how a function behaves.

The presence of a \(\frac{d^{3} y}{d x^{3}}\) in our equation signifies that the highest derivative is at the third level, which determines its order.

Higher-order derivatives, such as the third-order, often appear in complex real-world contexts, such as:

• In physics, when analyzing motion such as jerk, which is the derivative of acceleration.
• In technology, when implementing controls in robotics to smooth out movement patterns.

For the ODE in our exercise, recognizing the third-order derivative tells us not only about the classification and complexity of the equation but also about potential methods of solution.

In summary, understanding this third-order context provides insight into how to structure effective solutions and interpret real-world situations that the equation models.