lecture_17

# lecture_17 - MA 36600 LECTURE NOTES WEDNESDAY FEBRUARY 25...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MA 36600 LECTURE NOTES: WEDNESDAY, FEBRUARY 25 Complex Roots Euler’s Formula. We showed in the previous lecture that e it = cos t + i sin t. This is known as Euler’s Formula . In general, if we write r = λ + iμ in terms of real numbers λ and μ , we have the expression e rt = e λt · e iμt = e λt (cos μt + i sin μt ) = e λt cos μt + ie λt sin μt. In fact, because we have the Taylor Series expansion e rt = ∞ X k =0 r k k ! t k = ⇒ d dt e rt = r e rt for any complex number r . Hence we can always make sense of the function y ( t ) = e rt as a solution to a homogeneous linear differential equation with constant coefficients. Example. We explain how to find the general solution to the differential equation y 00 + 9 y = 0 . We guess a solution in the form y ( t ) = e rt so that we have the characteristic equation r 2 + 9 = 0 = ⇒ r = ± 3 i. Hence the general solution is the function y ( t ) = a 1 e 3 i + a 2 e- 3 i = a 1 (cos3 t + i sin3 t ) + a 2 (cos3 t- i sin3 t ) = c 1 cos3 t + c 2 sin3 t in terms of the constants c 1 = a 1 + a 2 and c 2 = ia 1- ia 2 . Review. Consider the constant coefficient second order differential equation ay 00 + by + cy = 0 . Recall that it has the associated characteristic equation ar 2 + br + c = 0 . We assume that b 2- 4 ac < 0 so that we have complex roots. We may write these complex roots as r 1 = λ + iμ r 2 = λ- iμ in terms of λ =- b 2 a , μ = p | b 2- 4 ac | 2 a . We saw in the previous lecture that we have the expression e r 1 t = e λt · e iμt = e λt (cos μt + i sin μt ) = e λt cos μt + ie λt sin μt ; e r 2 t = e λt · e- iμt = e λt (cos μt- i sin μt ) = e λt cos μt- ie λt sin μt ....
View Full Document

{[ snackBarMessage ]}

### Page1 / 4

lecture_17 - MA 36600 LECTURE NOTES WEDNESDAY FEBRUARY 25...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online