Solve ODE w/ Variation of Coefficient Technique: Complex Derivatives

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    Complex Derivatives
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Discussion Overview

The discussion revolves around solving an ordinary differential equation (ODE) using the variation of coefficients technique, specifically focusing on the derivatives of complex exponential functions, e^ix and e^-ix. Participants explore the differentiation process and seek clarification on applying these concepts in the context of ODEs.

Discussion Character

  • Technical explanation
  • Homework-related
  • Debate/contested

Main Points Raised

  • Some participants provide the first and second derivatives of e^ix and e^-ix, stating that the first derivative is i*e^ix and -i*e^-ix, respectively, and the second derivative is -e^ix and -e^-ix.
  • One participant suggests treating i as a constant during differentiation, emphasizing that the differentiation rules still apply.
  • Another participant notes that if the variable is complex (e.g., z), one must be more careful in the differentiation process.
  • Several participants express uncertainty about the application of the Cauchy-Riemann equations and whether to express the exponential function in terms of sine and cosine.
  • Some participants share personal experiences of struggling with similar derivative problems and recommend resources like Schaum's outlines for review.
  • There are repeated requests for tips or references to assist with understanding the derivatives of complex exponentials.

Areas of Agreement / Disagreement

Participants generally agree on the derivatives of e^ix and e^-ix, but there is no consensus on the best approach to apply these derivatives in the context of ODEs or the relevance of the Cauchy-Riemann equations. The discussion remains unresolved regarding the application of these concepts in more complex scenarios.

Contextual Notes

Some participants express uncertainty about the assumptions underlying the differentiation of complex exponentials and the implications of treating i as a constant. There are also unresolved questions about the appropriate context for applying the Cauchy-Riemann equations.

Logik
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I have to solve an ODE with variation of coefficient technique. It's pretty easy but I have no clue what is the first and second derivative of e^ix and e^-ix.
 
Last edited:
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If i had been a real number, what would the first and second derivatives have been then?
 
e^ix
first
i*e^ix
second
i^2*e^ix

e^-ix
first
-i*e^-ix
second
i^2*e^-ix

p.s. I've read about the Cauchy-Riemann equation but just not sure how to apply it... should I split the exponential in a sin and a cos?
p.s.s. There are probably rules, like exponential function are always derivable or something but I'm not fallowing any complex variables class right now so any insight is appreciated...
 
Last edited:
Logik said:
e^ix
first
i*e^ix
second
i^2*e^ix

e^-ix
first
-i*e^-ix
second
i^2*e^-ix

EXACTLY!
And that is precisely what holds when "i" is a complex/imaginary number as well! :smile:
 
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When dealing with these things, forget i is anything, just remember its a constant. Then after the actual differentiation, you can remember what it is.
 
Yeah. If

[tex]\exp(ix),\,\,\,x\in \mathbb{R},[/tex]

(which is what it looks like you have) then it's what the above two said. But if you have

[tex]\exp(iz),\,\,\,z\in \mathbb{Z},[/tex]

you need to be more careful. Let us know if that is indeed what you have.
 
what are you doing in a de course ifm you do not know the derivative of e^z?
 
I'm doing the same derivative problem & i was wondering if you could give any tips on how to solve the derivative of e^ix? I would really appreciate it. A good reference website, anything assistance at all.
 
bgbaby said:
I'm doing the same derivative problem & i was wondering if you could give any tips on how to solve the derivative of e^ix? I would really appreciate it. A good reference website, anything assistance at all.
That is exactly what has been answered in each of these responses. For any constant, a, the derivative of [itex]e^{ax}[/itex] is [itex]ae^{ax}[/itex].

That is a result of the very basic fact that the derivative of [itex]e^x[/itex] is [itex]e^x[/itex] (world's easiest derivative!) and the chain rule.
 
  • #10
bgbaby said:
I'm doing the same derivative problem & i was wondering if you could give any tips on how to solve the derivative of e^ix? I would really appreciate it. A good reference website, anything assistance at all.

[tex]\frac{d}{dx}(e^{jx})=je^{jx}[/tex]
[tex]\frac{d^2}{dx^2}(e^{jx})=-e^{jx}[/tex]
 
  • #11
Actually I myself was once in an ode course when I had forgot the derivative of e^x. My solution was to go get a Schaum's outline series of ode and do a lot of problems and review my $$$ off.
 

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