What are the rules for a differential equation to be linear?

A differential equation is considered linear when the variables and their derivatives only appear in a specific way. The core rules focus on how the variables and their derivatives interact within the equation. According to the provided reference, here are the essential criteria:

Rules for Linear Differential Equations

To determine if a differential equation is linear, we must analyze the terms containing the dependent variable and its derivatives. Here's a breakdown of the key requirements:

• Multiplication by Constants: The dependent variable and its derivatives must only be multiplied by constants or functions of the independent variable. They cannot be multiplied by other dependent variables or their derivatives.
• First Power: Each dependent variable and its derivatives must appear to the first power. This means there are no squared terms, square roots, or other non-linear power relationships involving the dependent variable or its derivatives.

In other words, these two conditions must be satisfied for each term involving the dependent variable or its derivatives:

1. No dependent variable multiplication: The dependent variable or its derivatives can not be multiplied by other dependent variables, derivatives of those dependent variables.
2. No non-linear terms: There cannot be squared terms, square roots, etc.

Here’s a table summarizing the rules:

Feature	Linear Differential Equation	Non-Linear Differential Equation
Variables/Derivatives	Multiplied by constants or functions of the independent variable only.	Multiplied by other dependent variables or derivatives.
Power	Appear to the first power only.	Appear to any other power, such as squared or inside a square root.

Examples of Linear Differential Equations

Let's explore some examples to solidify understanding:

• Example 1: dy/dx + 2y = x
  • This is linear because dy/dx and y are both to the first power, and they are multiplied by constants (1 and 2, respectively) or by a function of x (which is the independent variable).
• Example 2: d²y/dx² + (sin x) dy/dx + 5y = cos x
  • This is linear because all the derivatives and the dependent variable y are multiplied by either constants or functions of the independent variable x, and they all appear to the first power.
• Example 3: x^2(d^3y/dx^3) + x(dy/dx) + y = 0
  • This is linear because even though the derivatives are multiplied by functions of x, they do satisfy the two requirements to be considered linear.

Examples of Non-Linear Differential Equations

Here are some examples of non-linear differential equations and the reasons why they are not linear:

• Example 1: dy/dx + y^2 = x
  • This is non-linear because the dependent variable y appears as a square.
• Example 2: d²y/dx² + y * dy/dx = 0
  • This is non-linear because y is multiplied by its derivative dy/dx.
• Example 3: sin(dy/dx) + y = x
  • This is non-linear because dy/dx is part of the sin function.

Practical Insights

Understanding the distinction between linear and non-linear differential equations is crucial because:

• Solving Techniques: Linear equations often have well-established methods for finding solutions, whereas non-linear equations are frequently more challenging and may not have analytical solutions.
• Modeling: Many real-world systems can be approximated by linear models, which simplifies analysis and allows for useful predictions. Non-linear models, while sometimes more accurate, are also more difficult to work with.

In summary, the linearity of a differential equation hinges on whether the dependent variable and its derivatives are multiplied only by constants or functions of the independent variable, and if these terms appear to the first power. These rules ensure that the differential equation is well-behaved and can be analyzed using simpler methods.