(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Substitute [tex] p = \frac{dx}{dt} [/tex] to solve [tex]x\prime\prime + \omega^2x = 0 [/tex]

2. Relevant equations

[tex] \frac{dp}{dx} = v + x\frac{dv}{dx} [/tex]

[tex] v = \frac{p}{x}[/tex]

3. The attempt at a solution

[tex] p = \frac{dx}{dt}, \frac{dp}{dt} = \frac{d^2x}{dt^2} [/tex]

[tex] \frac{dp}{dt} = \frac{dp}{dx}\frac{dx}{dt} = \frac{dp}{dx}p [/tex]

[tex] \frac{dp}{dx} + \frac{\omega^2x}{p} = 0 [/tex]

[tex] v + x\frac{dv}{dx} = \frac{-\omega^2}{v} [/tex]

[tex] \frac{-v}{\omega^2 + v^2}dv = \frac{1}{x} dx [/tex]

[tex] \frac{-1}{2}ln(\omega^2 + v^2) = ln|x| + C [/tex]

[tex] \frac{1}{\sqrt{\omega^2 + v^2}} = x + C [/tex]

I get tripped up here and I'm not sure how to go forward, with regards to all the various substitutions I've made! I see the beginnings of an integral involving trigonometric substitution, so I hope I may be on the right track. A hint would be much appreciated.

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# Homework Help: Second order differential equation via substitution

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