A differential equation is classified as linear based on two key criteria related to the dependent variable and its derivatives.
Criteria for Linearity
A differential equation is considered linear if it satisfies the following conditions:
- Degree of 1: The equation is of the first degree in the dependent variable and its derivatives. This means that the dependent variable and its derivatives only appear to the power of 1.
- Coefficients: The coefficients in the equation are functions of only the independent variable.
Understanding the Criteria
Essentially, a linear differential equation has the form:
a_n(x) * d^n(y)/dx^n + a_{n-1}(x) * d^{n-1}(y)/dx^{n-1} + ... + a_1(x) * dy/dx + a_0(x) * y = f(x)Where:
yis the dependent variable.xis the independent variable.a_n(x), a_{n-1}(x), ..., a_0(x)are coefficients that are functions ofxonly.f(x)is a function ofxonly.
Examples of Linear vs. Nonlinear Differential Equations
Here's a table summarizing the key differences with examples:
| Feature | Linear Differential Equation | Nonlinear Differential Equation |
|---|---|---|
| Dependent Variable & Derivatives | Appears only to the first power (degree 1) | Appears with powers other than 1 (e.g., squared, cubed) |
| Terms | No transcendental functions (e.g., sin, cos, exp) of the dependent variable or its derivatives. | Contains transcendental functions of the dependent variable or its derivatives (e.g., sin(y), e^(dy/dx)). |
| Products | No products of the dependent variable and its derivatives with each other. | Contains products of the dependent variable and its derivatives (e.g., y * dy/dx). |
| Example | dy/dx + 2y = x^2 |
dy/dx + y^2 = x |
| Example | x^2 * d^2(y)/dx^2 + x * dy/dx + y = cos(x) |
d^2(y)/dx^2 + sin(y) = 0 (Pendulum Equation) |
Example Explained: Pendulum Equation
The reference mentions the equation representing the motion of a circular pendulum: d^2(y)/dx^2 + Sin[y[x]] = 0. This is nonlinear because Sin[y[x]] is not a linear function of y[x]. The sine function of the dependent variable makes the equation nonlinear.
Key Takeaways
- Linear differential equations are easier to solve than nonlinear ones.
- Linearity allows for the use of superposition principles.
- Many real-world phenomena are modeled by nonlinear differential equations, but linear approximations are often used for simplicity.
