Auxiliary Equation with Imaginary Roots

  1. cronxeh

    cronxeh 1,232
    Gold Member

    I was curious about what class would cover those types of Linear DE w Constant Coeff, particularly Hyperbolic Functions and exp z type of things. I remember my lecturer said back in Intro DE that we only covered first 2 types of Auxiliary Equations - real distinct roots and real repeated ones, but not the imaginary roots because they are 'out of the scope of this course' :frown:
     
  2. jcsd
  3. Pyrrhus

    Pyrrhus 2,276
    Homework Helper

    On any Differential Equations course, or ODE course.
     
  4. Zurtex

    Zurtex 1,123
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    It's almost exactly the same, but some times you use the different form by the identity:

    [tex]e^{x + iy} \equiv e^x \left( \sin y + i \cos y \right)[/tex]
     
  5. saltydog

    saltydog 1,593
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    Cronxeh, when you have imaginary roots to an equation, then the solution is of the form:

    [tex]y(x)=c_1e^{(a+bi)x}+c_2e^{(a-bi)x}[/tex]

    (and other more complex expressions for repeated complex roots)

    You can convert this using Euler's equation:

    [tex]e^{(a+bi)x}=e^{ax}\left(Cos(bx)+iSin(bx)\right)[/tex]

    to an expression containing exp's, sin's and cos's. Still have the i though. Can you separate the converted expression into a real part and imaginary part like:

    [tex]y(x)=r(x)+iv(x)[/tex]

    If you do, you'll get something like:

    [tex]i(c_1-c_2)[/tex]

    as a coefficient on the imaginary part. But that's a constant, call it [itex]k_2[/itex]. Now the solution is:

    [tex]y(x)=k_1r(x)+k_2v(x)[/tex]

    See how that works?
     
  6. cronxeh

    cronxeh 1,232
    Gold Member

  7. HallsofIvy

    HallsofIvy 40,548
    Staff Emeritus
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    " I remember my lecturer said back in Intro DE that we only covered first 2 types of Auxiliary Equations - real distinct roots and real repeated ones, but not the imaginary roots because they are 'out of the scope of this course' "

    That's a pretty weak D.E. course- even for "Intro". I would hope that your school also has a higher level D.E. course.
     
  8. cronxeh

    cronxeh 1,232
    Gold Member

    we cover imaginary roots but not from cauchy-euler equations, and this course is only 2 credits and lasts half a semester anyway
     
  9. Zurtex

    Zurtex 1,123
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    We covered exactly the same in Calc A at University. Excpet is was all done in 30 miniuites and our Tutor is so slow at ocvering stuff it missed out loads. I'm so glad I did Further Maths at A Level.
     
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