Question

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

Short answer

The order of the differential equation is 1.

Step-by-step solution

Identify Derivatives

Examine the given differential equation: \( \left( y^{\prime} \right)^{2} = y^{\prime} + 2y \). The derivative \( y^{\prime} \) is present but no higher derivatives are seen.

Determine the Highest Derivative

In \( \left( y^{\prime} \right)^{2} = y^{\prime} + 2y \), the highest derivative present is \( y^{\prime} \), which is the first derivative of \( y \).

Conclusion

The order of a differential equation is determined by the highest derivative present. Therefore, the order of this differential equation is 1.

Key concepts

Order of Differential Equations

The concept of the "order" of a differential equation is crucial to understanding its complexity and solution behavior. Simply put, the order of a differential equation is determined by the highest derivative present within the equation.
For example, in the equation \( (y')^2 = y' + 2y \), the highest derivative we encounter is \( y' \), which signifies the first derivative.
This tells us that there are no second derivatives \( y'' \), third derivatives \( y''' \), or any higher derivatives. Therefore, the order of this particular differential equation is 1.

• A first-order differential equation involves only the first derivative \( y' \).
• A second-order differential equation includes the second derivative \( y'' \), and so on.

Understanding the order helps identify the types of initial conditions that might be necessary and the nature of potential solutions.

First Derivative

The first derivative, often denoted as \( y' \), is a fundamental tool in calculus and differential equations.
It represents the rate of change of a function with respect to its independent variable, often time or space.
In the equation \( (y')^2 = y' + 2y \), the first derivative \( y' \) is utilized on both sides.
Here are some important points to understand about the first derivative:

• It gives us the slope of the tangent line to the graph of a function.
• A positive \( y' \) indicates that the function is increasing, while a negative \( y' \) means it is decreasing.
• In physical terms, it can represent speed, velocity, or other rates of change.

This equation showing \( y' \) squared highlights the ability of the first derivative to interact in non-linear ways in different equations.

Higher Derivatives

While the given equation, \( (y')^2 = y' + 2y \), is a first-order equation, many situations require higher derivatives.
Higher derivatives are simply derivatives of derivatives. For instance, the second derivative \( y'' \) is the derivative of \( y' \).
Even if they don't appear in every equation, understanding higher derivatives is important for grasping dynamic behaviors of systems.

• Second derivatives \( y'' \) review concavity of a function, showing where it curves upwards or downwards.
• Third derivatives \( y''' \) and beyond can be important in more complex analyses, like evaluating the rate of change of acceleration.

They provide greater insights into the motion, shape, and behavior of functions and are essential in fields like physics and engineering.