Question

Determine the order of the following differential equations.\(\frac{d y}{d x}+\frac{d^{2} y}{d x^{2}}=3 x^{4}\)

Short answer

The order is 2.

Step-by-step solution

Identify the Derivatives

First, identify all the derivative terms present in the differential equation. In the equation \( \frac{d y}{d x}+\frac{d^{2} y}{d x^{2}}=3 x^{4} \), we have two derivative terms: \( \frac{d y}{d x} \) and \( \frac{d^{2} y}{d x^{2}} \).

Determine the Highest Order Derivative

Once the derivatives are identified, determine the highest order derivative present in the equation. In this equation, the term \( \frac{d^{2} y}{d x^{2}} \) represents the second derivative and is of higher order than the first derivative \( \frac{d y}{d x} \).

Conclude the Order of the Equation

The order of a differential equation is given by the highest order derivative present in the equation. Therefore, since the highest derivative is \( \frac{d^{2} y}{d x^{2}} \), the order of this differential equation is 2.

Key concepts

Understanding the Order of Differential Equations

The order of a differential equation is a crucial concept that aids in understanding its complexity and the nature of solutions it might have. The order refers to the highest derivative present in the equation. For instance, if you come across an equation like \( \frac{d y}{d x}+\frac{d^{2} y}{d x^{2}}=3 x^{4} \), you need to determine which derivative has the highest order.
To do this, identify all derivative terms. Here, you have \( \frac{d y}{d x} \) (the first derivative) and \( \frac{d^{2} y}{d x^{2}} \) (the second derivative).
Since \( \frac{d^{2} y}{d x^{2}} \) is a second derivative, this equation is called a second-order differential equation.

• First identify all derivatives.
• Find the derivative with the highest order.
• The order of the differential equation is the same as this highest order derivative.

This understanding is foundational, as the order of the differential equation influences the method of solving it and the form of its solution.

Delving into Derivatives

Derivatives are at the heart of calculus and play a significant role in differential equations. They express how a function changes as its input changes. In mathematical terms, a derivative measures the rate at which a quantity changes with respect to another.
The first derivative of a function \( y \), represented as \( \frac{d y}{d x} \), shows how \( y \) changes with a slight change in \( x \). This is analogous to the slope of a curve at any point.
Higher-order derivatives, like the second derivative \( \frac{d^{2} y}{d x^{2}} \), can show more complex relationships.

• First derivatives often represent velocity or speed in physical contexts.
• Second derivatives can illustrate acceleration, or the rate of change of the rate of change.
• Understanding derivatives is vital for solving differential equations.

In differential equations, you may encounter multiple derivatives all interacting within a single equation, helping describe complex phenomena.

Exploring Second-Order Differential Equations

Second-order differential equations appear frequently in various scientific disciplines, from physics to engineering. These equations involve second derivatives, making them more complex than first-order ones, as more detailed behavior or change is captured.
Consider an equation like \( \frac{d^{2} y}{d x^{2}} + \frac{d y}{d x} = 3x^4 \). Here, the presence of \( \frac{d^{2} y}{d x^{2}} \) indicates that it is a second-order differential equation.

• They can model oscillatory systems, such as springs or circuits.
• Approaches to solve second-order differential equations include characteristic equations and undetermined coefficients.

Understanding how these equations work provides insight into the natural and technological processes, highlighting the important role of second derivatives in capturing dynamic behaviors.