Question

Identify the dependent and independent variables of the following differential equations. Give the order of the equations and state which are linear. (a) \(\frac{\mathrm{d} y}{\mathrm{~d} x}+9 y=0\) (b) \(\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)\left(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\right)+3 \frac{\mathrm{d} y}{\mathrm{~d} x}=0\) (c) \(\frac{\mathrm{d}^{3} x}{\mathrm{~d} t^{3}}+5 \frac{\mathrm{d} x}{\mathrm{~d} t}=\sin x\)

Short answer

Question: Identify the dependent and independent variables, the order of the equation, and whether the equation is linear or not for the following differential equations: (a) \( \frac{\mathrm{d} y}{\mathrm{~d} x}+9 y=0\), (b) \(\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)\left(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\right)+3 \frac{\mathrm{d} y}{\mathrm{~d} x}=0\), and (c) \(\frac{\mathrm{d}^{3} x}{\mathrm{~d} t^{3}}+5 \frac{\mathrm{d} x}{\mathrm{~d} t}=\sin x\). Answer: (a) Dependent variable: \(y\), Independent variable: \(x\), Order: 1, Linearity: Linear (b) Dependent variable: \(y\), Independent variable: \(x\), Order: 2, Linearity: Nonlinear (c) Dependent variable: \(x\), Independent variable: \(t\), Order: 3, Linearity: Linear

Step-by-step solution

Dependent variable

In the first equation, the dependent variable is \(y\) as it depends on the independent variable, \(x\).

Independent variable

In this equation, the independent variable is \(x\). #Part (a): Order of Differential Equation#

Order of Equation

Since the highest derivative appearing in the equation is the first derivative, the equation is of order 1. #Part (a): Linearity#

Linearity

The given equation is linear because it involves only linear terms with respect to the dependent variable y and its derivatives. #Part (b): Dependent & Independent Variables#

Dependent variable

In the second equation, the dependent variable is \(y\) as it depends on the independent variable, \(x\).

Independent variable

In this equation, the independent variable is \(x\). #Part (b): Order of Differential Equation#

Order of Equation

Since the highest derivative appearing in the equation is the second derivative \( \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\), the equation is of order 2. #Part (b): Linearity#

Linearity

The given equation is nonlinear because it involves a product of the first and second derivatives of the dependent variable y. #Part (c): Dependent & Independent Variables#

Dependent variable

In the third equation, the dependent variable is \(x\) as it depends on the independent variable, \(t\).

Independent variable

In this equation, the independent variable is \(t\). #Part (c): Order of Differential Equation#

Order of Equation

Since the highest derivative appearing in the equation is the third derivative \(\frac{\mathrm{d}^{3} x}{\mathrm{~d} t^{3}}\), the equation is of order 3. #Part (c): Linearity#

Linearity

The given equation is linear because it involves only linear terms with respect to the dependent variable x and its derivatives.

Key concepts

Dependent and Independent Variables

When we discuss differential equations, it's important to identify both dependent and independent variables. In an equation, the dependent variable is the one that depends on the other variable. Essentially, its value is determined by the independent variable.
For example, consider Part (a) of our exercise: - This equation is written as \(\frac{\mathrm{d} y}{\mathrm{~d} x}+9 y=0\). - Here, the function \(y\) is the dependent variable, indicating that its value changes based upon the variable \(x\), which is the independent variable.
Understanding which variable is dependent and which is independent helps in analyzing how changes affect the system being modeled by the differential equation. In Part (b), \(y\) again acts as the dependent variable and \(x\) is the independent variable in the equation \(\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)\left(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\right)+3 \frac{\mathrm{d} y}{\mathrm{~d} x}=0\).
In Part (c), the dependent variable is \(x\) because it is expressed in terms of \(t\), making \(t\) the independent variable in the equation \(\frac{\mathrm{d}^{3} x}{\mathrm{~d} t^{3}}+5 \frac{\mathrm{d} x}{\mathrm{~d} t}=\sin x\).
Recognizing these variables is the stepping stone to analyzing and solving differential equations.

Order of Differential Equation

The 'order' of a differential equation is a fundamental concept that refers to the highest derivative present in the equation.
The order indicates how many times a function is differentiated. Let's dive into the examples given:

• For Part (a), the equation \(\frac{\mathrm{d} y}{\mathrm{~d} x}+9 y=0\) involves the first derivative \(\frac{\mathrm{d} y}{\mathrm{~d} x}\). Thus, it is a first-order differential equation.
• In Part (b), the equation \(\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)\left(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\right)+3 \frac{\mathrm{d} y}{\mathrm{~d} x}=0\) includes the second derivative \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\), making it a second-order differential equation.
• Lastly, in Part (c), the equation \(\frac{\mathrm{d}^{3} x}{\mathrm{~d} t^{3}}+5 \frac{\mathrm{d} x}{\mathrm{~d} t}=\sin x\) contains the third derivative \(\frac{\mathrm{d}^{3} x}{\mathrm{~d} t^{3}}\), which classifies it as a third-order differential equation.

Understanding the order is vital as it indicates the nature and complexity of the solutions. Higher-order equations typically describe more complex systems and require more initial conditions to solve them.

Linearity of Differential Equations

In the world of differential equations, linearity is an important property that dictates how the solution behaves and how it can be solved.
A differential equation is linear if it is a linear combination of the dependent variable and its derivatives—that is no powers or products of these terms are present.
Let's break down the examples:

• In Part (a), the equation \(\frac{\mathrm{d} y}{\mathrm{~d} x}+9 y=0\) is linear. It has no products or powers of y and its derivatives, adhering to the linearity criterion.
• Part (b) presents the equation \(\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)\left(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\right)+3 \frac{\mathrm{d} y}{\mathrm{~d} x}=0\), which is nonlinear. This is because the presence of the product \(\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)\left(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\right)\) makes it a nonlinear equation.
• Lastly, in Part (c), the equation \(\frac{\mathrm{d}^{3} x}{\mathrm{~d} t^{3}}+5 \frac{\mathrm{d} x}{\mathrm{~d} t}=\sin x\) is also nonlinear due to the sine function \(\sin x\), which introduces nonlinearity.

Recognizing whether a differential equation is linear or not helps in choosing appropriate solution methods. Linear equations are often more straightforward to solve and have well-defined techniques for obtaining solutions.