Problem 123 - Calculus 3

Question

Classify the differential equation. Determine the order, whether it is linear and, if linear, whether the differential equation is homogeneous or nonhomogeneous. If the equation is second-order homogeneous and linear, find the characteristic equation. $$\left(\frac{d y}{d t}\right)^{2}+y y^{\prime}=1$$

Step-by-Step Solution

Answer

The equation is a first-order, nonlinear differential equation.

Step 1: Identify Equation Type

We begin by examining the given differential equation: \(\left(\frac{dy}{dt}\right)^{2} + y \frac{dy}{dt} = 1\). Observe that the equation involves derivatives of \(y\) with respect to \(t\).

Step 2: Determine the Order

The order of a differential equation is defined as the highest derivative that appears in the equation. Here, the highest derivative is \(\frac{dy}{dt}\), which is the first derivative. Therefore, the order of the differential equation is 1.

Step 3: Check for Linearity

For an equation to be linear, the dependent variable (\(y\) and its derivatives) should appear at most to the first power and not be multiplied together. In this equation, \(\left(\frac{dy}{dt}\right)^{2}\) has the derivative squared, and \(y \frac{dy}{dt}\) has the product of \(y\) and its derivative, making it nonlinear.

Step 4: Determine Homogeneity (If Applicable)

Since the equation is identified as nonlinear in Step 3, determining homogeneity is not applicable here. Homogeneity is only examined in linear differential equations.

Key Concepts

Order of Differential Equation

In the realm of differential equations, the order is one of the first things we identify. The order refers to the highest derivative present in the equation.

• A derivative represents the rate at which a quantity changes.
• The notation \( \frac{dy}{dt} \) signifies the first derivative of \( y \) with respect to \( t \).
• The order tells us the complexity of the relationship the equation depicts, based on how many rates of change it includes.

In the given exercise, we looked at the equation \( \left(\frac{dy}{dt}\right)^{2} + y \frac{dy}{dt} = 1 \). We observe that the highest derivative is \( \frac{dy}{dt} \), a first derivative. Thus, the equation is of first order.
First-order differential equations are often simpler and govern systems that revert to steady states or have simple dynamics.

Linear vs Nonlinear Differential Equations

Differential equations can be classified as linear or nonlinear, which affects how they are analyzed and solved. In general:

• A linear differential equation implies that each term is either a constant or a product of a constant with the first power of the dependent variable or its derivatives.
• Nonlinear equations, on the other hand, can have derivatives raised to powers other than one, products of derivatives, or functions of the derivatives.

In our exercise, the equation \( \left(\frac{dy}{dt}\right)^{2} + y \frac{dy}{dt} = 1 \) has a term \( \left(\frac{dy}{dt}\right)^{2} \), which indicates a derivative squared. It also has \( y \frac{dy}{dt} \), which is a product of \( y \) and its derivative, making the equation nonlinear. Nonlinear equations can describe more complex systems, like chaotic dynamics or certain biological processes.

Homogeneous vs Nonhomogeneous

Homogeneity is a concept primarily associated with linear differential equations. We do not typically apply it to nonlinear cases.

• A linear differential equation is homogeneous if all terms depend on the dependent variable or its derivatives, and there's no constant or standalone function on the other side of the equal sign.
• If there is a constant or a function unmatched by a counterpart using the dependent variable, the equation is nonhomogeneous.

Since our equation \( \left(\frac{dy}{dt}\right)^{2} + y \frac{dy}{dt} = 1 \) is nonlinear, the criterion of homogeneity does not apply. In contrast, if we did have a linear case, determining whether it is homogeneous helps in identifying the nature of solutions, as homogeneous equations typically describe undriven systems.