Dealing with complex roots and more on independence

Dealing with complex roots and more on independence -...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
Dealing with complex roots and more on independence
Background image of page 1

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

View Full DocumentRight Arrow Icon
Complex roots When the roots of the auxiliary equation for the differential equation ay’’+by’+cy=0 ar 2 +br+c=0 are complex ( b 2 4ac<0 ) , they come as a complex conjugate pair r= α ±i β In that case we have (complex) solutions y h =C 1 e ( α +i β )t +C 2 e ( α‐ i β )t =e α t (C 1 e i β t +C 2 e i β t )
Background image of page 2
Complex roots y h =e α t (C 1 e i β t +C 2 e i β t ) =e α t (C 1 ( cos β t +isin β t) +C 2 ( cos β t isin β t) ) =e α t ((C 1 +C 2 ) cos β t + (iC 1 iC 2 ) sin β t =e α t (D 1 cos β t +D 2 sin β t ) since the C’s were arbitrary constants, we can reformulate and use D’s. This form has the advantage of producing real values when D’s are real.
Background image of page 3

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

View Full DocumentRight Arrow Icon
In the case with one real root, (b 2 =4ac ) we can seek a second independent solution in the form y=ue rt . In this case, we know that r= b/2a Plugging in to the differential equation we get a(u”+2ru’+r
Background image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 04/10/2008 for the course MATH 254 taught by Professor Indik during the Spring '08 term at Arizona.

Page1 / 9

Dealing with complex roots and more on independence -...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online