1
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The order of an ordinary differential equation is the number of primes in the highest order derivative
2
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In this case, x is the highest order derivative, and so the order of the ODE is 3
3
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The ODE in Problem 2 can be converted to a system of n first-order ODEs like this
4
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First you get the highest-order derivative by itself on the left hand side of the equation
5
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Then you define yourself some helper variables
6
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If you have a second-order ODE, you just need to define yourself one helper variable
7
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If I had a third-order ODE, Id have to define myself two helper variables
8
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Theyd look something like x = v, v = a
9
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But here, I just need one
10
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Then the next step is to rewrite the first equation using the helper variable, like this
11
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Now, it doesnt matter what you call your helper variable; you could choose whatever you want here
12
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It is important to make sure that there are no derivatives on the right-hand side of these equal signs
13
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Thats actually why we define the helper variables, so that we can rewrite everything on the right-hand side without any derivatives
14
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So this set of equations is not the right answer, because this would be the set of equations that would correspond to a third-order ODE; thats quite different
15
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This set of equations is almost right, except for this term right here
16
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Remember, we want no derivatives on the right hand sides of the equal signs
17
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This is the right answer
18
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This one has a sign problem, and this one doesnt have the helper variable in it, so its not complete