Differential Equations Classification Flashcards

Study Notes

Classification of Differential Equations

Independent Variable: Represents the variable typically displayed on the bottom of a function or equation; it's the input value that can be manipulated.
Dependent Variable: Appears at the top of the function or equation; it is the output that relies on the independent variable.
Linear Equation: Defined as an equation where dependent variables do not involve trigonometric functions, are not squared, and are not multiplied by their own derivatives.
Ordinary Differential Equation (ODE): A type of differential equation that involves only one independent variable.
Partial Differential Equation (PDE): Involves two or more independent variables, expanding the complexity beyond ODEs.
Order of a Differential Equation: Determined by the highest derivative present; for example, a first-order equation has the first derivative, while a fourth-order equation has up to the fourth derivative.
Autonomous Differential Equation: Features an independent variable that is not explicitly present in the equation, leading to simpler analysis in some cases.
Normal Form of a Differential Equation: Specifically formatted as dy/dx = [expression], highlighting the relationship between the derivative and other terms.
Standard Form of a Differential Equation: Structured as dy/dx + y = [expression], a common representation that includes the dependent variable and its derivative.
Homogeneous Solution: Represents a differential equation defined by the form dy/dx + y = 0, indicating solutions that cancel out the non-homogeneous part.