Taylor Series and Differential Equations

Taylor Series and Differential Equations

  • Taylor Series
    • Understand that a Taylor series is a representation of a function as an infinite sum of terms.
    • Know that these terms are calculated from the values of the function’s derivatives at a single point.
    • This single point in the function is often referred to as the “centre” and is denoted by ‘a’. The Taylor series of a real or complex-valued function f(x) that is infinitely differentiable at a real or complex number a is the power series: f(a) + f’(a)(x-a) + f’‘(a)(x-a)^2/2! + … .
    • Be aware that the nth derivative of the function is divided by n factorial (n!).
    • The Taylor series is a specific type of power series, more specifically, a Maclaurin series is a Taylor series expansion of a function about 0.
  • Differential Equations
    • Understand that a differential equation is an equation that involves derivatives of a function.
    • Remember that the order of a differential equation is the order of the highest derivative present.
    • Know that there are two types of differential equations: ordinary differential equations, which only have one variable; and partial differential equations, which consist of more than one independent variable.
    • Recall first order differential equations, where you can use an integrating factor to simplify the equation to a form where it can be integrated.
    • Be able to write down the general solution of a second order homogeneous linear differential equation, which is of the form ay’’ + by’ + cy = 0, where a, b and c are constants.
    • Understand how to apply boundary conditions to find a particular solution to a differential equation.
    • Be aware that the term “linear differential equation” means that the dependent variable and its derivatives are raised to the first power and multiplied by coefficients that typically depend on the independent variable.

Application of Taylor Series to Differential Equations

  • Understand that the Taylor Series is used to estimate solutions to differential equations.
  • Be aware that solutions are often approximated because exact solutions cannot always be found.
  • Understand how to use the Taylor Series to solve a differential equation step by step, expanding the function in terms of derivatives at a point.
  • Realize the difference between an exact solution and an approximated solution derived from a finite number of terms from the Taylor series.