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Calculus Volume 2

Key Concepts

Calculus Volume 2Key Concepts

Key Concepts

4.1 Basics of Differential Equations

  • A differential equation is an equation involving a function y=f(x)y=f(x) and one or more of its derivatives. A solution is a function y=f(x)y=f(x) that satisfies the differential equation when ff and its derivatives are substituted into the equation.
  • The order of a differential equation is the highest order of any derivative of the unknown function that appears in the equation.
  • A differential equation coupled with an initial value is called an initial-value problem. To solve an initial-value problem, first find the general solution to the differential equation, then determine the value of the constant. Initial-value problems have many applications in science and engineering.

4.2 Direction Fields and Numerical Methods

  • A direction field is a mathematical object used to graphically represent solutions to a first-order differential equation.
  • Euler’s Method is a numerical technique that can be used to approximate solutions to a differential equation.

4.3 Separable Equations

  • A separable differential equation is any equation that can be written in the form y′=f(x)g(y).y′=f(x)g(y).
  • The method of separation of variables is used to find the general solution to a separable differential equation.

4.4 The Logistic Equation

  • When studying population functions, different assumptions—such as exponential growth, logistic growth, or threshold population—lead to different rates of growth.
  • The logistic differential equation incorporates the concept of a carrying capacity. This value is a limiting value on the population for any given environment.
  • The logistic differential equation can be solved for any positive growth rate, initial population, and carrying capacity.

4.5 First-Order Linear Equations

  • Any first-order linear differential equation can be written in the form y′+p(x)y=q(x).y′+p(x)y=q(x).
  • We can use a five-step problem-solving strategy for solving a first-order linear differential equation that may or may not include an initial value.
  • Applications of first-order linear differential equations include determining motion of a rising or falling object with air resistance and finding current in an electrical circuit.
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