What is the differential equation which is linear?

A linear differential equation is one that can be written in a specific form, exhibiting linearity in the dependent variable and its derivatives.

Understanding Linear Differential Equations

According to the provided reference, a first-order linear differential equation typically takes the form:

dy/dx + Py = Q

Where:

• y is the dependent variable (a function of x).
• x is the independent variable.
• dy/dx is the first derivative of y with respect to x.
• P and Q are either numeric constants or functions of x only. They cannot be functions of y.

Characteristics of Linear Differential Equations

Here's a breakdown of what makes a differential equation linear:

• Dependent Variable and its Derivatives are of the First Degree: This means that y and dy/dx appear only to the first power. There are no terms like y^2, (dy/dx)^3, or sqrt(y).
• No Products of the Dependent Variable and its Derivatives: Terms like y * (dy/dx) are not allowed.
• Coefficients Depend Only on the Independent Variable: The functions P and Q can only depend on x.

Examples

Here are a few examples to illustrate what linear and non-linear differential equations look like:

Linear Differential Equations:

• dy/dx + 2y = x
• dy/dx + (sin x)y = x^2
• dy/dx + y = e^x

Non-Linear Differential Equations:

• dy/dx + y^2 = x (Non-linear because of y^2)
• dy/dx + y(dy/dx) = x (Non-linear because of y * (dy/dx))
• dy/dx + sin(y) = x (Non-linear because of sin(y))

General Form for Higher-Order Linear Differential Equations

While the reference focuses on first-order equations, linear differential equations can also be of higher order. The general form of an nth-order linear differential equation is:

a_n(x) dⁿy/dxⁿ + a_n-1(x) d^n-1y/dx^n-1 + ... + a₁(x) dy/dx + a₀(x)y = f(x)

Where:

• a_n(x), a_n-1(x), ..., a₁(x), a₀(x), and f(x) are functions of x only.
• dⁿy/dxⁿ represents the nth derivative of y with respect to x.

Importance of Linearity

Linear differential equations are easier to solve than non-linear ones. There are well-established methods for finding their solutions, which makes them important in many areas of science and engineering. The principle of superposition applies to linear differential equations, meaning that if you have two solutions, their sum is also a solution.