Second order differential equations Watch

gangsta316
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I encountered this question from The Muon yesterday.

http://www.thestudentroom.co.uk/show...36&postcount=1

I managed to solve it and got the simple harmonic equation in the form shown. I previously tried to derive this simple harmonic motion equation by using the complementary function but I couldn't get it into the right form, and the fact that there were i's (imaginary unit) all over the place confused me. I found the method shown here to be much better. When can this method here be used for a second order ODE question? Is it only for questions where the roots of the complementary function are purely imaginary? Or does it only happen to work out in this question?
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generalebriety
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Suppose ax'' + bx' + cx = 0 (where primes denote differentiation with respect to time). Then av.dv/dx + bv + cx = 0. When b = 0 or c = 0, this is separable and easy. When neither of them is zero, it's obviously harder.
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DFranklin
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(Original post by gangsta316)
I encountered this question from The Muon yesterday.

http://www.thestudentroom.co.uk/show...36&postcount=1

I managed to solve it and got the simple harmonic equation in the form shown. I previously tried to derive this simple harmonic motion equation by using the complementary function but I couldn't get it into the right form, and the fact that there were i's (imaginary unit) all over the place confused me.
As far as trying to solve this using complementary functions: your problem was that you were trying to use the "wrong" complementary functions.

In solving a problem you would normally the form of the CF based on the roots and whether or not they are complex. So looking at this problem I would immediately think "roots are +/- w, so CF = A cos wt + B sin wt". Which is obviously much easier than the alternative method - it's not even 1 full line of working. (The advantage of the alternative method is that it actually derives the answer. With the CF method you are essentially assuming the answer (using a "magic formula") rather than deriving it, Usually we just want the answer though, so we don't care).

So I would strongly advise you to learn the different forms of the CF (depending on whether the roots are real or complex) rather than give up on the approach. You'll be at a big disadvantage if you don't know how to solve these problems using CFs.
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