Question

What is the order of the differential equation? $$ \left(y^{\prime \prime \prime}\right)^{4}-\frac{t^{2}}{\left(y^{\prime}\right)^{4}+4}=0 $$

Short answer

Answer: The order of the given differential equation is 3.

Step-by-step solution

Identify the highest order derivative

In the given equation, we have derivatives with respect to t: \(y'\) (first derivative), and \(y^{\prime \prime \prime}\) (third derivative). The highest order derivative present is the third derivative, \(y^{\prime \prime \prime}\).

Determine the order of the differential equation

The order of a differential equation is the order of the highest-order derivative present in the equation. Since the highest order derivative we found is \(y^{\prime \prime \prime}\) (third derivative), the order of the given differential equation is 3.

Key concepts

Understanding Differential Equations

Differential equations are a cornerstone of mathematics and engineering, encapsulating the relationship between a function and its derivatives. Fundamentally, these equations involve an unknown function, typically represented as a variable (like y), and one or more derivatives of that function with respect to another variable (such as t).

They play a pivotal role in modeling the dynamics of various systems, from the simple motion of a falling object affected by gravity to the complex changes in ecosystems over time. A differential equation's solution offers a formula that can predict the behavior of the system under examination.

The Basic Structure

A simple differential equation may look like this: \(\frac{dy}{dt} = g(t)\), where \(g(t)\) is some function involving t. The expression \(\frac{dy}{dt}\) is the first derivative of y with respect to t, indicating the rate of change of y.

Deciphering the Third Derivative

The third derivative in calculus is the rate of change of the rate of change of the rate of change. Sound complex? It's certainly a concept that can take a moment to wrap your head around, yet it's an essential tool for understanding motion and curvature within mathematical models.

When considering physical applications, the third derivative of position with respect to time, for instance, gives us an idea of how the acceleration of an object is changing—a concept known as jerk in physics. Going beyond velocity and acceleration, jerk helps predict the future path of motion with more detail.

Mathematical Notation

In notation, the third derivative of a function y with respect to t is usually written as \(y^{\backprime\backprime\backprime}(t) = \frac{d^3y}{dt^3}\), making it clear we're dealing with the third level of differentiation from the original function.

Identifying the Highest Order Derivative

The highest order derivative in a differential equation signifies the derivative with the greatest number of differentiations. It's the key to determining the 'order' of the equation, a fundamental property in the study of differential equations.

As seen in our example, the presence of the third derivative \(y^{\backprime\backprime\backprime}\) means we're dealing with an equation of at least the third order. No other derivative in the equation, no matter how complex it may appear, will increase the order once the third derivative is present.

To enhance understanding and ensure complex problems are approachable, identify the highest order derivative first. This step is not merely procedural but offers insight into the potential solutions and the behavior of the differential equation itself.