Second Order Equations

Complex Roots of the Characteristic Equation

Recall the Characteristic Equation

Remember, to solve

We find roots of the characteristic equation

FindWrite the general solution:

and

Recall the Characteristic Equation

We find roots of the characteristic equation

In general, using quadratic formula

This is fine if the discriminant

Recall the Characteristic Equation

This is fine if the discriminant

But what if

?Then

is imaginary!This results in imaginary or complex

roots.

ExampleHas the characteristic

equation

And roots

Which are:

But what to do?

Well, let’s just plug it in

That’s pretty ugly!

How do we get rid of those imaginary numbers?

Euler’s Magic FormulaRemember Euler’s Magic

Formula

So for Our Problem

We can rewrite:

Euler’s Magic FormulaSo for Our

Problem

We can rewrite:

which gives us

Euler’s Magic FormulaSo for Our

Problem

We can rewrite:

which gives us

Euler’s Magic FormulaSo for Our

Problem

We can rewrite:

which gives us

Euler’s Magic FormulaSo for Our

Problem

We can rewrite:

which gives us

Euler’s Magic FormulaSo for Our

Problem

We can rewrite:

which gives us

Euler’s Magic FormulaSo for Our

Problem

We can rewrite:

which gives us

Euler’s Magic FormulaSo for Our

Problem

We can rewrite:

which gives us

Euler’s Magic FormulaSo for Our

Problem

We can rewrite:

which gives us

Euler’s Magic FormulaSo for Our

Problem

We can rewrite:

which gives us

Euler’s Magic FormulaSo for Our

Problem

We can rewrite:

which gives us

General SolutionSo for Our Problem

The general solution is

But this part is imaginary!We want real solutions….

General SolutionSo for Our Problem

The general solution is

Not actually a problem

General SolutionSo for Our Problem

The general solution is

Two ways to think about why:Way #1:Is Some Arbitrary

(Possibly Complex) ConstantIs Some Other Arbitrary

(Possibly Complex) Constant

General SolutionSo for Our Problem

The general solution is

Two ways to think about why:Way #1:So

General SolutionSo for Our Problem

The general solution is

Two ways to think about why:Way #1:So

and it turns out

andare real for any real initial

conditions

General SolutionSo for Our Problem

The general solution is

Two ways to think about why:Way #2:satisfies the homogeneous

equationsatisfies the homogeneous equationWronskian

of and

is

So form

aandfundamental set of

solutions.

General SolutionSo for Our Problem

The general solution is

General CaseReturning to the General

Case

We find roots of the characteristic equation

If the discriminant

These terms

are imaginary

General CaseReturning to the General

Case

We find roots of the characteristic equation

If the discriminant

And the same

General CaseReturning to the General

Case

We find roots of the characteristic equation

If the discriminant

And the same

General CaseReturning to the General

Case

We find roots of the characteristic equation

Differ by only the minus

sign:called a

“Conjugate Pair”

If the discriminant

General CaseReturning to the General

Case

Insert into standard form

Apply Euler’s Formula

General CaseReturning to the General

Case

Insert into standard form

Rearrange and collect terms

So to solve

If the characteristic function

Has Complex Roots

Solution takes the form

Proceed as usual for Homogeneous Constant Coefficients

To find particular solutions, plug in initial conditions and solve.

Summary

• (Last Friday) - If characteristic function has distinct real roots ->

• (Today) - If characteristic function has complex roots ->

• (Next Monday) - What if characteristic function only has one root?

To Solve

Questions?