1. Second Order Equations Complex Roots of the Characteristic Equation

2. Recall the Characteristic Equation Remember, to solve We find roots of the characteristic equation Find Write the general solution: and

3. Recall the Characteristic Equation We find roots of the characteristic equation In general, using quadratic formula This is fine if the discriminant

4. Recall the Characteristic Equation This is fine if the discriminant But what if ? The n is imaginary! This results in imaginary or complex roots.

5. Example Has the characteristic equation And roots Which are:

6. But what to do? Well, let’s just plug it in That’s pretty ugly! How do we get rid of those imaginary numbers?

7. Euler’s Magic Formula Remember Euler’s Magic Formula So for Our Problem We can rewrite:

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

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

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

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

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

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

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

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

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

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

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

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

20. 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

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

22. 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

23. 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.

24. General Solution So for Our Problem The general solution is

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

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

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

28. 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

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

30. General Case Returning to the General Case Insert into standard form Rearrange and collect terms

31. 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.

32. 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

33. Questions?