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Separable Differential Equations definitions

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  • Separable Differential Equation
    An equation where the derivative can be written as a product of a function of one variable and a function of another.
  • Variable Separation
    The process of rearranging an equation so that all terms involving one variable are on one side with its differential.
  • Integration
    A step used after separating variables to find the general or particular solution to a differential equation.
  • Constant of Integration
    An arbitrary value added after integrating, determined by initial conditions in specific problems.
  • Initial Condition
    A given value for the function at a specific point, used to find the particular solution of a differential equation.
  • General Solution
    A solution to a differential equation that contains an arbitrary constant, representing a family of functions.
  • Particular Solution
    A specific solution to a differential equation found by applying initial conditions to the general solution.
  • Exponential Growth
    A process where a quantity increases at a rate proportional to its current value, modeled by a positive growth constant.
  • Exponential Decay
    A process where a quantity decreases at a rate proportional to its current value, modeled by a negative growth constant.
  • Growth Constant
    A parameter indicating the rate of exponential change; positive for growth, negative for decay.
  • Newton's Law of Cooling
    A model describing temperature change as proportional to the difference between an object's temperature and its environment.
  • Surrounding Medium
    The environment whose temperature influences the rate of temperature change in Newton's law of cooling.
  • Implicit Solution
    A solution where the dependent variable cannot be isolated explicitly, often left in a combined form.
  • Natural Logarithm
    A mathematical function, often appearing in solutions after integrating expressions involving 1 over a variable.
  • Differential Equation
    An equation involving derivatives that describes how a quantity changes with respect to another variable.