Question

Determine the order of the following differential equations.\(y^{\prime}=y^{\prime \prime}+3 t^{2}\)

Short answer

The order of the differential equation is 2.

Step-by-step solution

Identify the Differential Equation

First, let's write the given differential equation to identify the derivatives involved. The equation provided is \( y' = y'' + 3t^2 \).

Examine the Terms

Examine each term in the equation. The equation includes \( y' \) (first derivative of \( y \)) and \( y'' \) (second derivative of \( y \)). The \( 3t^2 \) term is not a derivative of \( y \).

Determine the Order of the Differential Equation

The order of a differential equation is the highest order of derivative present in the equation. Here, the highest order derivative is \( y'' \), which is a second derivative.

Key concepts

Order of Differential Equations

The order of a differential equation is determined by identifying the highest derivative within the equation. In any differential equation, derivatives reflect how a function changes over different variables. The key is to pinpoint the derivative with the highest order, as this dictates the complexity and nature of the equation. In the given equation, we have both a first and a second derivative: \( y' = y'' + 3t^2 \).
In simple terms:

• First derivative, denoted as \( y' \), indicates the rate of change of \( y \) with respect to another variable such as \( t \).
• Second derivative, denoted as \( y'' \), shows how the rate of change itself is changing.

Among these, the second derivative \( y'' \) is of the highest order. Hence, the order of the differential equation is 2. This order is essential in understanding the behavior of solutions and the type of methods needed to solve the equation.

First Derivative

The first derivative, represented as \( y' \), is a foundational concept in calculus and a fundamental aspect of differential equations. It measures how a function \( y \) changes concerning another variable, such as time \( t \).
Here’s a simple way to grasp it:

• It tells how quickly or slowly \( y \) is increasing or decreasing as \( t \) changes.
• In practical terms, if you think of \( y \) as distance and \( t \) as time, the first derivative \( y' \) informs us of the speed or velocity.

In our specific equation \( y' = y'' + 3t^2 \), \( y' \) is on the left side indicating it is being directly equated with other terms involving derivatives and functions of \( t \). Understanding the first derivative helps in predicting the immediate direction and magnitude of change in the function under analysis.

Second Derivative

The second derivative, noted as \( y'' \), goes a step further from the first derivative. While the first derivative describes the rate of change of a function, the second derivative shows how that rate itself changes over time or another variable.
More insights include:

• It is crucial in determining the curvature or concavity of the function \( y \).
• If \( y'' > 0 \), the function is concave up, meaning it opens upwards like a cup, indicating acceleration in terms of motion.
• If \( y'' < 0 \), conversely, it is concave down, signifying deceleration.

In the equation \( y' = y'' + 3t^2 \), the presence of \( y'' \) highlights its role as the highest order derivative, making it pivotal in understanding the behavior of the system described by this equation over time as \( t \) changes. Second derivatives are powerful tools in physics and engineering as they relate to acceleration and forces acting on objects.