Definition

1. Noun:
   • An equation that relates a function with its derivatives: A differential equation is a mathematical equation that involves an unknown function and one or more of its derivatives. It describes a relationship between the function and its rates of change.
   • A tool for modeling change: Differential equations are fundamental in mathematically describing phenomena where quantities change, such as motion, growth, decay, or wave propagation.

Usage Examples

• Noun:
  • Newton's second law of motion is expressed as a differential equation.
  • Solving this differential equation will tell us how the population grows over time.
  • The course covers techniques for solving linear differential equations.

Advanced Usage

• "Ordinary differential equation (ODE)": A differential equation involving a function of a single independent variable and its derivatives.
  • The motion of a pendulum is modeled by an ordinary differential equation.
• "Partial differential equation (PDE)": A differential equation involving a function of multiple independent variables and its partial derivatives.
  • The heat equation is a famous partial differential equation in physics.
• "To satisfy a differential equation": For a function to be a solution, it must make the equation true when substituted into it.
  • The exponential function satisfies the simple differential equation dy/dx = y.

Variants and Related Words

• Derivative (n): The instantaneous rate of change of a function, a core component of differential equations.
• Integral (n): Related concept; solving a differential equation often involves integration.
• Differential (adj/n): Pertaining to derivatives or infinitesimal differences.

Synonyms

• DiffEQ (n, informal): A common abbreviation used in academic settings.
• Equation involving derivatives: A descriptive synonym.

Related Phrases

• "Boundary value problem": A differential equation accompanied by constraints (boundary conditions) specifying the solution at the boundaries of the domain.
• "Initial value problem": A differential equation accompanied by conditions specifying the solution at a starting point.

Related Idioms

• "Separation of variables": A standard method for solving certain types of differential equations.
• "Characteristic equation": An algebraic equation derived from a linear differential equation to find its solution.

Noun

1. an equation containing differentials of a function