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2.2: Classification of Differential Equations

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Recall that a differential equation is an equation (has an equal sign) that involves derivatives. Just as biologists have a classification system for life, mathematicians have a classification system for differential equations. We can place all differential equation into two types: ordinary differential equation and partial differential equations.

Examples 2.2.1

d2ydx2+dydx=3xsiny

is an ordinary differential equation since it does not contain partial derivatives. While

yt+xyx=x+txt

is a partial differential equation, since y is a function of the two variables x and t and partial derivatives are present.

In this course we will focus on only ordinary differential equations.

Order

Another way of classifying differential equations is by order. Any ordinary differential equation can be written in the form

F(x,y,y,y,...,y(0))=0

by setting everything equal to zero. The order of a differential equation is the highest derivative that appears in the above equation.

Examples 2.2.2

d2ydx2+dydx=3xsiny

is a second order differential equation, since a second derivative appears in the equation.

3y4yx3y+exyy=0

is a third order differential equation.

Once we have written a differential equation in the form

F(x,y,y,y,...,y(n))=0

we can talk about whether a differential equation is linear or not. We say that the differential equation above is a linear differential equation if

Fy(i)y(j)=0

for all i and j. Any linear ordinary differential equation of degree n can be written as

a0(x)y(n)+a1(x)y(n1)+...+an1(x)y+an(x)y=g(x).

Examples 2.2.3

3x2y+2ln(x)y+exy=3xcosx

is a second order linear ordinary differential equation.

4yyx3y+cosy=e2x

is not a linear differential equation because of the 4yy and the cosy terms.

Nonlinear differential equations are often very difficult or impossible to solve. One approach getting around this difficulty is to linearize the differential equation.

Example 2.2.4: Linearization

y+2y+ey=x

is nonlinear because of the ey term. However, the Taylor expansion of the exponetial function

ey=1+y+y22+y36+...

can be approximated by the first two terms

ey1+y.

We instead solve the much easier linear differential equation

y+2y+1+y=x.

We say that a function f(x) is a solution to a differential equation if plugging in f(x) into the equation makes the equation equal.

Example 2.2.5

Show that

f(x)=x+e2x

is a solution to

y2y=2.

Solution

Taking derivatives:

f(x)=1+2e2x,f(x)=4e2x.

Now plug in to get

4e2x2(1+2e2x)=4e2x24e2x=2.

Hence it is a solution.

Two questions that will be asking repeatedly of a differential equation course are

  1. Does there exist a solution to the differential equation?
  2. Is the solution given unique?

In the example above, the answer to the first question is yes since we verified that

f(x)=x+e2x

is a solution. However, the answer to the second question is no. It can be verified that

s(x)=4+x

is also a solution.

Larry Green (Lake Tahoe Community College)

  • Integrated by Justin Marshall.


This page titled 2.2: Classification of Differential Equations is shared under a not declared license and was authored, remixed, and/or curated by Larry Green.

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