Transforming Equations for the Simple Harmonic Oscillator

  • Thread starter noranne
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In summary, the discussion revolves around finding solutions to a simple harmonic oscillator problem, specifically the expressions for C and del in terms of A and B, and for D and E in terms of A and B. The conversation also touches on the use of Euler's Formula and the clarification of a typo in one of the equations.
  • #1
noranne
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I guess this is maybe more algebra than calculus, but it stems from a calculus problem, so I'll stick it here.

The problem is:

In the case of the simple harmonic oscillator the solution [to the EOM] may be written at least 3 ways

x(t) = Acos(wt) + Bsin(wt)
= Ccos(wt + del)
= De^(iwt) + Ee^(1wt)

Express C and del in terms of A and B. Express D and E in terms of A and B.

What I've got:

I got the first part, C and del, but I can't figure out how to find D and E. It seems relatively straightforward, I put the A/B eq as the LHS and the D/E eq on the RHS and just applied Euler's Formula to the D term. But I can't figure out how to get the E term in terms of cos and sin.

Any help please?
 
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  • #2
Hint: A constant that has the imaginary unit in it is still a constant.
 
  • #3
Yes, I know, but that still doesn't help me get E*e^(wt) in terms of cos and sin.
 
  • #4
D*exp(iwt)+E*exp(wt) is NOT a solution to a simple harmonic oscillator problem in the region of the universe I'm used to. Are you sure you don't mean D*exp(iwt)+E*exp(-iwt)? You certainly can decompose exp(wt) into sin and cos. It's exp(wt)=exp(i(wt/i))=cos(wt/i)+i*sin(wt/i). But if w and t are real, those sin and cos aren't the oscillatory functions you'd expect.
 
  • #5
Yeah, one of my friends just told me "Didn't you get Prof's email? That's a typo!"

Gr. I KNEW that I wasn't doing it wrong.

Thanks!
 

1. What is the purpose of transforming equations?

Transforming equations involves manipulating mathematical expressions in order to simplify or solve for a specific variable. It allows us to make complex equations more manageable and can provide insight into the relationship between different variables.

2. What are the different methods of transforming equations?

Some common methods of transforming equations include substitution, elimination, and rearranging terms. These methods involve applying algebraic rules and operations to an equation to change its form and make it easier to solve.

3. Can equations be transformed without changing their solutions?

Yes, equations can be transformed without changing their solutions as long as the same operations are applied to both sides of the equation. This is known as the "golden rule" of equation solving.

4. How do I know when an equation has been fully transformed?

An equation is fully transformed when it is in its simplest form and the desired variable is isolated on one side of the equals sign. This means that no further operations can be applied to the equation.

5. Are there any common mistakes to avoid when transforming equations?

Some common mistakes to avoid when transforming equations include forgetting to perform the same operation on both sides of the equation, making a sign error when distributing or combining terms, and accidentally dropping or adding terms. It is important to carefully check each step and make sure the equation remains balanced throughout the transformation process.

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