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Math 555: Differential Equations

1.3 Classification of Differential Equations

1.3.1 Ordinary and Partial Differential Equations

Differential Equations are classified by several properties. The broadest classification determines the type of differential equation by how many independent variables the equations contain. If the differential equation contains only one independent variable it is called ordinary . All of the differential equations studied in this course are ordinary differential equations ( ODEs ). If the differential equation involves more than one independent variable the differential equation is call a partial differential equation ( PDEs ). Examples of partial differential equations are the Laplace equation

$$u_{xx} + u_{yy} + u_{zz} = 0,$$
the heat equation

$$ u_{xx} + u_{yy} + u_{zz} - \kappa u_t = 0, $$
and the wave equation

$$ u_{xx} + u_{yy} + u_{zz} - \dfrac{1}{c^2}u_{tt} = 0. $$
In each of these differential equations the three space variables $x$, $y$, and $z$ are independent variables and time $t$ is an independent variable.

While we usually use $x$ and $t$ for independent variables, that is not necessary. We will also sometimes treat our independent and dependent variables as inputs to a single function. We will need to differentiate these single functions with respect to both inputs. These derivatives are also called partial derivatives and we will discuss partial derivatives in chapter 3 when we need them.

1.3.2 General Form of an Ordinary Differential Equation

The most general way to express an ordinary differential equation writes the highest order derivative of the dependent variable as a function of the independent variable, the dependent variable and all of the lower order derivatives of the independent variable.

  1. $\dfrac{dy}{dt} = ay + b$

  2. $\dfrac{dy}{dx} = \cos(ax) - \sin(by)$

  3. $\dfrac{d^2y}{dt^2} = \dfrac{dy}{dt} + \sin(\omega t)$

  4. $y'''(t) = -3y'' + 4y' + 2y - 5x$

All of these differential equations are ordinary differential equations because there is only one independent variable. The order of the differential equation is degree of the highest derivative that appears in the differential equation. The degree of each ordinary differential equation in this section is

  1. 1
  2. 1
  3. 2
  4. 3

1.3.3 More General Differential Equations

A differential equation does not need to be written in general form.

  1. $\dfrac{dy}{dt} - ay = b$

  2. $\dfrac{dy}{dx} + \sin(by) = \cos(ax)$

  3. $\dfrac{d^2y}{dt^2} - \dfrac{dy}{dt} = \sin(\omega t)$

  4. $y'''(t) + 3y'' - 4y' - 2y = 5x$

Each of the differential equations from section 1.3.2 is re-written above.

1.3.4 Homogeneous Ordinary Differential Equations

A differential equation is called homogeneous if every term has the dependent variable or one of its derivatives as a factor.

  1. $3y'' - 2y' + y = 0$

  2. $y''' - xy'' + x^2y' - y^2 = 0$

  3. $e^ty' + \sin(t)y = 0$

  4. $\dfrac{d^2\theta}{dt^2} + \dfrac{g}{L}\sin(\theta) = 0$

  5. $t^2y'' + 4ty' + 2y = 0$

All of these differential equations are homogeneous ordinary differential equations.

  1. $3y'' - 2y' + y - e^{t/2} = 0$

  2. $y''' - xy'' + x^2y' - y^2 = x^2\sin(2x)$

  3. $e^ty' + \sin(t)y = \cos(3t)$

  4. $\dfrac{d^2\theta}{dt^2} + \dfrac{g}{L}\sin(\theta) = \cos(\omega t)$

  5. $t^2y'' + 4ty' + 2y - 5 = 0$

This list is made up of non-homogeneous ordinary differential equations. They are not homogeneous because at least one term, even a constant term, does not have the dependent variable or one of its derivatives as a factor. These terms are

  1. $e^{t/2}$
  2. $x^2\sin(2x)$
  3. $\cos(3t)$
  4. $\cos(\omega t)$
  5. $-5$

1.3.5 Linear Ordinary Differential Equation

We will start with a naive definition of a linear differential equation for chapters 1 and 2, and then we will improve our understanding in chapter 3. For chapter 1 and chapter 2 we look at the order of the independent variable in every term it appears. The best way to explain this is to examine some examples.

$$y''$$
is a linear term of order two because we are computing two derivatives.

$$y$$
is a linear term of order zero but it still has the dependent variable as a factor.

$$y\cdot y''$$
is a term of order two but unlike the previous two terms, it is not a linear term because we have two factors of the dependent variable in this term.

$$t^3y''$$
is a linear term of order two; the order is two because we are expressing the second derivative. The term is linear because the dependent variable appears only once as a factor. The $t^3$ factor and the expression of any function of the independent variable is irrelevant to determining if the term is linear or nonlinear.

$$y^2$$
is a nonlinear term of order zero.

$$t^2y$$ is a linear term of order zero.

$$\sin(y)$$ is a nonlinear term of order zero. It is nonlinear because the definition of $\sin(y)$ we learned in calculus is that

$$\sin(y) = \displaystyle\sum_{n=0}^{\infty} (-1)^n\cdot\dfrac{y^{2n+1}}{(2n+1)!},$$
which we can think of as a polynomial that a term of every odd positive degree. However

$$\sin(t^2)$$ is a linear term because it has no dependent variable as a factor. Hence

  1. $\dfrac{d^2y}{dt^2} + \sin(y) = \sin(t)$ is a nonlinear ODE

  2. $\dfrac{d^2y}{dt^2} + y = \sin(t)$ is a linear ODE

  3. $\dfrac{d^2y}{dt^2} + y^2 = \sin(t)$ is a nonlinear ODE

  4. $\dfrac{d^2y}{dt^2} + t^2y = \sin(t)$ is a linear ODE

  5. $\dfrac{d^2y}{dt^2} + y\cdot y' = \sin(t)$ is a nonlinear ODE

1.3.6 Systems of Equations

Finally we defined an ordinary differential equation to have only one independent variable but we did not restrict the number of dependent variables.

$$\begin{array}{rcl} \dfrac{dx}{dt} & = & ax - \alpha xy \\ \\ \dfrac{dy}{dt} & = & -cy + \gamma xy \end{array}$$
is a differential equation with one independent variable $t$ and two dependent variables $x$ and $y$. Notice that it is a nonlinear differential equation because of the terms $\alpha xy$ and $\gamma xy$. Notice also that we have written the differential equation as a system of two first order differential equations; one that expresses $\frac{dx}{dt}$ and the other expresses $\frac{dy}{dt}$. We can express them together as a vector-valued differential equation

$$ \begin{bmatrix} x \\ y \end{bmatrix}^{'} = \begin{bmatrix} a & -\alpha x \\ \gamma y & -c \end{bmatrix} \cdot \begin{bmatrix} x \\ y \end{bmatrix} $$

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