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Basics of Differential Equations definitions

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  • Differential Equation
    An equation involving a function and its derivatives, possibly of any order, often relating y and its derivatives to other expressions.
  • Order
    The highest derivative present in an equation, used to classify the complexity of a differential equation.
  • Linearity
    A property determined by whether the function and its derivatives appear only to the first power, are not multiplied together, and are not inside other functions.
  • Nonlinearity
    A characteristic where the function or its derivatives are multiplied together, raised to powers other than one, or appear as arguments of other functions.
  • Solution
    A function that, when substituted along with its derivatives into a differential equation, makes the equation true.
  • General Solution
    A family of functions containing an arbitrary constant, representing all possible solutions to a differential equation before initial values are applied.
  • Particular Solution
    A specific function from the general solution, found by applying initial conditions to determine the constant.
  • Initial Condition
    A value or set of values specifying the function and possibly its derivatives at a particular point, used to find a unique solution.
  • Antiderivative
    A function whose derivative is the given function, used to solve differential equations by integration.
  • Constant of Integration
    An arbitrary constant added after integrating, representing the infinite set of possible antiderivatives.
  • Implicit Form
    An equation where the function is not isolated, sometimes requiring implicit differentiation to verify solutions.
  • Verification
    The process of substituting a proposed function and its derivatives into a differential equation to check if the equation holds.
  • Dependent Variable
    The function whose values depend on the independent variable, commonly denoted as y in differential equations.
  • Family of Functions
    A set of functions differing by a constant, all satisfying a given differential equation.
  • Argument
    The input of a function, such as y or its derivatives appearing inside another function like sine or exponential.