1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Complex Numbers and Constants of Integration

  1. Dec 6, 2015 #1

    squelch

    User Avatar
    Gold Member

    1. The problem statement, all variables and given/known data

    Suppose that the characteristic equation to a second order, linear, homogeneous differential equation with constant coefficients yielded two complex roots:
    [tex]\begin{array}{l}
    {\lambda _1} = a + bi\\
    {\lambda _2} = a - bi
    \end{array}[/tex]
    This would yield a general solution of:
    [tex]y = {C_1}{e^{(a + bi)x}} + {C_2}{e^{(a - bi)x}}[/tex]

    I would like to prove that this is equal to the expression:
    [tex]y = {C_1}{e^{ax}}\sin (bx) + {C_2}{e^{ax}}\cos (bx)[/tex]

    2. Relevant equations

    Euler's identity:
    [tex]{e^{ix}} = \cos (x) + i\sin (x)[/tex]

    3. The attempt at a solution

    At the end of the proof, I am left with the expression:
    [tex]y = i{C_1}{e^{ax}}\sin (bx) + {C_2}{e^{ax}}\cos (bx)[/tex]

    Can ##i## be "rolled up" into the constant of integration ##C_1## and the whole thing just defined as a single, undetermined constant?
     
  2. jcsd
  3. Dec 6, 2015 #2

    Samy_A

    User Avatar
    Science Advisor
    Homework Helper

    If ##C_1## and ##C_2## are the same constants as in ##y = {C_1}{e^{(a + bi)x}} + {C_2}{e^{(a - bi)x}}##, I don't think this is correct...
    ... and if they are differrent constants, why not "roll up" the ##i## into ##C_1##?
     
  4. Dec 6, 2015 #3

    squelch

    User Avatar
    Gold Member

    I'm just trying to derive a textbook definition, in case the derivation is required on an exam.
    At a point in the derivation, coming from the original equation, ##{e^{ax}}\cos (bx)[{C_1} + {C_2}] + i{e^{ax}}\sin (bx)[{C_1} - {C_2}]##.
    I combined the constants into a new ##C_1## and ##C_2##, mostly to match the textbook equation. I suppose it'd be more clear (and proper) to call them ##C_1 '## and ##C_2 '##
    I don't see a reason why I wouldn't be able to, but if it's a legal operation or not is my question. Call it a sanity check.
     
  5. Dec 6, 2015 #4

    Samy_A

    User Avatar
    Science Advisor
    Homework Helper

    Yes, you can do that. As the ##C##'s and ##C'##'s are arbitrary complex numbers anyway, there is absolutely no reason why you couldn't do it.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Complex Numbers and Constants of Integration
Loading...