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Auxiliary Equation with Imaginary Roots 
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#1
Jun2705, 07:12 PM

PF Gold
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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'



#3
Jun2705, 07:34 PM

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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] 


#4
Jun2705, 07:46 PM

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Auxiliary Equation with Imaginary Roots
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^{(abi)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_1c_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? 


#5
Jun2705, 09:24 PM

PF Gold
P: 1,236

Ah thanks. I didnt have time before but now that I'm home I did some digging and found those functions covered in this course:
http://www.wellesley.edu/Math/Math20...work/hwk6.html I'm taking Complex Variables in Fall, guess we'll be covering those then 


#6
Jun2805, 09:41 AM

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Thanks
PF Gold
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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. 


#7
Jun2805, 10:28 AM

PF Gold
P: 1,236

we cover imaginary roots but not from cauchyeuler equations, and this course is only 2 credits and lasts half a semester anyway



#8
Jun2805, 04:13 PM

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