Question

How do you recognize a linear homogeneous differential equation? Give an example and explain why it is linear and homogeneous.

Short answer

Answer: A linear homogeneous differential equation is an equation of the form: \[\sum_{i=0}^n a_i(x)y^{(i)}(x) = 0\] where \(a_i(x)\) are functions of \(x\), and \(y^{(i)}(x)\) are the \(i\)-th order derivatives of the function \(y(x)\). It is called "linear" because each term of the equation contains \(y(x)\), or its derivatives, raised to the power 1, and it is called "homogeneous" because the equation equals 0 on the right side. An example of a linear homogeneous differential equation is: \[\frac{d^2y}{dx^2} - 2\frac{dy}{dx} + y = 0\] This example is linear because each term containing \(y(x)\) or its derivatives are raised to the power of 1 and homogeneous since it equals 0 on the right side.

Step-by-step solution

Definition of Linear Homogeneous Differential Equation

A linear homogeneous differential equation is an equation of the form: \[\sum_{i=0}^n a_i(x)y^{(i)}(x) = 0\] where \(a_i(x)\) are functions of \(x\), and \(y^{(i)}(x)\) are the \(i\)-th order derivatives of the function \(y(x)\). It's called "linear" because each term of the equation contains \(y(x)\), or its derivatives, raised to the power 1. It's called "homogeneous" because the equation equals 0 on the right side.

Provide an Example

Let's consider the following differential equation: \[\frac{d^2y}{dx^2} - 2\frac{dy}{dx} + y = 0\]

Explain Why the Example is Linear

For this equation, we can see that all the terms containing \(y(x)\), or its derivatives, are raised to the power of 1: \[\begin{cases} \text{Term 1: } \frac{d^2y}{dx^2}\text{ has }y^{(2)} \text{ raised to the power of 1} \\ \text{Term 2: } -2\frac{dy}{dx}\text{ has }y^{(1)} \text{ raised to the power of 1} \\ \text{Term 3: } y\text{ has }y\text{ raised to the power of 1} \end{cases}\] Since each term in the equation is a linear function of the function \(y(x)\), or its derivatives, the given equation is linear.

Explain Why the Example is Homogeneous

Since the right side of the given equation is equal to 0: \[\frac{d^2y}{dx^2} - 2\frac{dy}{dx} + y = 0\] it means the equation is homogeneous. Therefore, the given example is indeed a linear homogeneous differential equation.