Classi cation of Ordinary Di erential Equations (ODE’s)

Classification of Ordinary Differential Equations (ODE's)

Math 181, Fall 2019 Instructor: Dr. Doreen De Leon

There are two major classes of ODE's, linear and nonlinear.

Linear Equations

Definition. A linear differential equation is a differential equation that can be written in

the form

pn(x)y(n) + pn-1(x)y(n-1) + ? ? ? + p1(x)y + p0(x)y = f (x),

(1)

where pn(x) = 0, and pj(x), j = 1, 2, . . . , n and f (x) are arbitrary functions of x.

If f (x) = 0, then (1) is said to be homogeneous; otherwise, (1) is nonhomogeneous.

Examples. The following equations are linear.

? t2y

+

t

3 2

y

=

0

? sin(x)y + tan(x)y = 12 ln(2x)

Exercise. Determine if each of the following equations is linear or nonlinear. If linear, determine if its is homogeneous or nonhomogeneous.

? y = y + 1 - linear, nonhomogeneous ? (y )2 = x + y - nonlinear ? y = ety - linear, homogeneous ? yy = x - nonlinear ? y = et + ey - nonlinear ? x3y = 3x2y + y - linear, homogeneous ? x2(y + x) = xy - y - linear, nonhomogeneous

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Order of a Differential Equation

Definition. The order of a differential equation is the order of the highest derivative of the dependent variable.

Example. In the exercise we just did, the first equation is second order, the second is first order, and the sixth is third order.

Initial Value Problems

Definition. An nth order differential equation

y(n) = f (x, y, y , . . . , y(n-1))

accompanied by values y(x0) = y0, y (x0) = y1, . . . , y(n-1)(x0) = yn-1, called initial conditions, is an initial value problem (IVP).

Note: The number of initial conditions needed for a unique solution to exist equals the order of the ODE.

Example. The population of bacteria in a petrie dish can be modeled by the separable

equation

dy = ky, y(0) = 20000,

dt

where k is the per capita growth rate. This is an IVP for which the general solution is given

by y(t) = cekt.

Substituting in the initial condition gives 20000 = y(0) = c, so y(t) = 20000ekt. In order to completely solve this problem, we would need more information, such as the population doubling time.

Autonomous Differential Equations

Definition. A differential equation is autonomous if it can be written in the form y(n) = f (y, y , . . . , y(n-1)).

Notes: 2

(1) In an autonomous equation, the independent variable (typically t) does not explicitly appear.

(2) While autonomous equations are generally nonlinear, this is not necessary. For example, y = y - 2 is autonomous and linear.

Autonomous First Order Equations

In general, given the autonomous first order equation

y = f (y),

(2)

if f (y) = 0 has real-valued solutions, say a0, a1, . . . , ap, then y(t) = a0, y(t) = a1, . . . , y(t) = an all solve (2). Conversely, if y(t) = a0, y(t) = a1, . . . , y(t) = ap all solve (2), then a0, a1, . . . , ap are all roots of f (y).

Note: Obviously, an equation need not be autonomous to have constant solutions. For example, the equation

y = x(y - 1), y(0) = 1

has the unique solution y(x) = 1.

Definition. Any solution of a first order differential equation

y = f (x, y)

for which y = 0 is called an equilibrium solution.

Solutions of Autonomous First Order Equations

Although it is not always possible to analytically solve an autonomous first order equation, it is possible to sketch solution curves using calculus. We will discuss qualitative solutions and analysis of equilibrium solutions in more detail for systems of equations later in the semester. Example. Consider the autonomous first order equation

y = y(y - 2)2.

(a) Find all equilibrium solutions. (b) Sketch representative solution curves for t 0, - < y < .

Solution.

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(a) To find the equilibrium solutions, set y = 0 and solve y(y - 2)2 = 0, giving y = 0, y = 2.

(b) To sketch the solution curves, first recall from calculus that if y > 0, then y is increasing and if y < 0, then y is decreasing. Also, a point of inflection xi is found where y (xi) = 0 and y (xi+) and y (xi-) have opposite signs. Since f (y) = y(y - 2)2, we have the following

y y 2: y > 0, so y is increasing. As y gets larger, y gets larger and the slope is

increasing.

A sketch of a few solution curves is below.

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Exercise. Consider the autonomous first order equation y = y2(y + 2).

(a) Find the equilibrium solutions. (b) Sketch representative solution curves for t 0, - < y < .

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