Section outline

  • Linear differential equation with constant coefficients

    Introduction:

    In this class, a linear differential equation with constant coefficients will be discussed.
    If the constant term is the zero function, then the differential equation is said to be homogeneous, as it is a homogeneous polynomial in the unknown function and its derivatives. ... A differential equation has constant coefficients if only constant functions appear as coefficients in the associated homogeneous equation.

    Homogeneous equation with constant coefficients
    A homogeneous linear differential equation has constant coefficients if it has the form 
    {\displaystyle a_{0}y+a_{1}y'+a_{2}y''+\cdots +a_{n}y^{(n)}=0}  
    where  a_1, \ldots, a_n are (real or complex) numbers. In other words, it has constant coefficients if it is defined by a linear operator with constant coefficients.

    Non-homogeneous equation with constant coefficients:
    A non-homogeneous equation of order n with constant coefficients may be written
    {\displaystyle y^{(n)}(x)+a_{1}y^{(n-1)}(x)+\cdots +a_{n-1}y'(x)+a_{n}y(x)=f(x),}
    where a_1, \ldots, a_n are real or complex numbers, f is a given function of x, and y is the unknown function (for sake of simplicity, "(x)" will be omitted in the following). There are several methods for solving such an equation. The best method depends on the nature of the function f that makes the equation non-homogeneous. If f is a linear combination of exponential and sinusoidal functions, then the exponential response formula may be used.

    Objectives:

    • Solve the homogeneous linear differential equations with constant coefficients.
    • Apply the method to solve the non-homogeneous linear differential equations with constant coefficients.
    • solve systems of linear differential equations.
    • determine the type of linear differential equation systems
    • Use the operator method to solve linear systems with constant coefficients.
    • Solve the linear systems in normal form.
    • Solve the homogeneous linear systems with constant coefficients.
    Lesson Plan:
    • Discussion from a lecture slide
    • Problem-solving
    • Group Work
    • Question and Answering
    • Home Work