Second Order Equations

So Far…We have been solving linear

and nonlinear first order

equations.Those days are

over.

Today, we will start examining

second order linear equations.

(Sort of)

Bad (or Good?) NewsHigher order equations are

MUCH more difficult to solve than

first order equations.

So we start simple, and work our way

up.

Linear equations are hard.Nonlinear equations

are crazy hard.

Linear Second Order Equations

The form of a second order linear equation:

Not simple enough!

Still not simple enough!

Homogenous

equation.

Linear Second Order Equations

The form of a second order linear equation:

Not simple enough!

Still not simple enough!

Linear Second Order Equations

The form of a second order linear equation:

Not simple enough!

Still not simple enough!

Okay, that’s a good place to start.

Second Order Linear Homogenous Equations with

Constant Coefficients

Side note:

We already know one specific solution to this

equation.

Equilibrium Solution!

Always check for equilibrium solutions!!!!

Second Order Linear Homogenous Equations with

Constant Coefficients

We will consider solutions of the form:

Why?

There is a reason.

We will discuss it later in the course.

Second Order Linear Homogenous Equations with

Constant Coefficients

We will consider solutions of the form:

Second Order Linear Homogenous Equations with

Constant Coefficients

We will consider solutions of the form:

Then:

Second Order Linear Homogenous Equations with

Constant Coefficients

We will consider solutions of the form:

Then:

Second Order Linear Homogenous Equations with

Constant Coefficients

We will consider solutions of the form:

Then:

Second Order Linear Homogenous Equations with

Constant Coefficients

We will consider solutions of the form:

Then:

Second Order Linear Homogenous Equations with

Constant Coefficients

Factoring out:

One of these two must be 0

Second Order Linear Homogenous Equations with

Constant Coefficients

Factoring out:

Can’t be 0

Second Order Linear Homogenous Equations with

Constant Coefficients

Factoring out:

Must be 0

Second Order Linear Homogenous Equations with

Constant Coefficients

Factoring out:

So

This is called the characteristic equation.

So to solve…

Find the characteristic equation

Find(Either factor or use the quadratic

equation)Write the general

solution:

The catch…Because there’s always a

catch….

FindTwo

solutions!

Three possibilities

or

So to solve…Find the characteristic

equation

FindWrite the general solution:

and

Specific SolutionsFor first order equations, need 1 initial

condition

For second order equations, need 2 initial conditions

For order N equations, need N initial conditions

Specific SolutionFor second order equations, need 2 initial

conditions

To find specific solution:

Plug in Initial Conditions:

Specific SolutionFor second order equations, need 2 initial

conditions

To find specific solution:

Plug in Initial Conditions:

Solve for:

and

So to solve…Find the characteristic

equationFindWrite the general solution:

and

Plug in Initial Conditions:

Solve for:

and

ExampleFind the characteristic

equationFind

and

and

ExampleFind the characteristic

equationFind

andan

dWrite the general solution:

Plug in Initial Conditions:

Solve for

andan

d

ExampleFind the characteristic

equationFind

andan

dWrite the general solution:

Plug in Initial Conditions:Solve

forandan

d

Summary• Higher Order Equations are Difficult To Solve

• Can solve Second Order Homogeneous Linear Equations with Constant Coefficients Using Characteristic Equations

• Need 2 Initial Conditions for Second Order Equations

• In general, need N initial conditions for Nth order equation

Questions?