Second Order Equations Complex Roots of the Characteristic Equation

Recall the Characteristic Equation Remember, to solve We find roots of the characteristic equation Find Write 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 ? The n is imaginary! This results in imaginary or complex roots.

Example Has 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 Formula Remember Euler’s Magic Formula So for Our Problem We can rewrite:

Euler’s Magic Formula So for Our Problem We can rewrite: which gives us

Euler’s Magic Formula So for Our Problem We can rewrite: which gives us

Euler’s Magic Formula So for Our Problem We can rewrite: which gives us

Euler’s Magic Formula So for Our Problem We can rewrite: which gives us

Euler’s Magic Formula So for Our Problem We can rewrite: which gives us

Euler’s Magic Formula So for Our Problem We can rewrite: which gives us

Euler’s Magic Formula So for Our Problem We can rewrite: which gives us

Euler’s Magic Formula So for Our Problem We can rewrite: which gives us

Euler’s Magic Formula So for Our Problem We can rewrite: which gives us

Euler’s Magic Formula So for Our Problem We can rewrite: which gives us

General Solution So for Our Problem The general solution is But this part is imaginary! We want real solutions….

General Solution So for Our Problem The general solution is Not actually a problem

General Solution So for Our Problem The general solution is Two ways to think about why: Way #1: Is Some Arbitrary (Possibly Complex) Constant Is Some Other Arbitrary (Possibly Complex) Constant

General Solution So for Our Problem The general solution is Two ways to think about why: Way #1: SoSo

General Solution So for Our Problem The general solution is Two ways to think about why: Way #1: SoSo and it turns out and are real for any real initial conditions

General Solution So for Our Problem The general solution is Two ways to think about why: Way #2: satisfies the homogeneous equation Wronskian ofand is Soform aand fundamental set of solutions.

General Solution So for Our Problem The general solution is

General Case Returning to the General Case We find roots of the characteristic equation If the discriminant These terms are imaginary

General Case Returning to the General Case We find roots of the characteristic equation If the discriminant And the same

General Case Returning to the General Case We find roots of the characteristic equation If the discriminant And the same

General Case Returning 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 Case Returning to the General Case Insert into standard form Apply Euler’s Formula

General Case Returning 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?