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# Remove arbitrary constants from the function $x = c_1 \cos wt + c_2 \sin wt$, where $c_1, c_2$ are arbitrary constants.

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Please help me. I've been trying to do and I can't resolve, because I'm confusing I don't know if  derive dt/dw or dw/dt. I know that I have to derive implicitly.

Thanks.

Note: Clay6 has edited the question for clarity.

Answer: $\large\frac{d^2x}{dt^2}$$+ w^2x = 0 Since the given function has two arbitrary constants, a second order differential equation has to be derived. Given x = c_1 \cos \;wt + c_2 \sin\;wt where c_1, c_2 are arbitrary constants that need to be eliminated. Here w is a parameter that cannot be eliminated. Differentiating, we get: \large\frac{dx}{dt}$$ = -w c_1\sin \; wt + wc_2 cos\;wt$
Differentiating again, we get: $\large\frac{d^2x}{dt^2}$$= -w^2c_1\cos \; wt - w^2 c_2 \sin wt = -w^2 (c_1\cos\;wt + c_2 \sin \;wt) Now, from the original equation, we can see that x = c_1 \cos \;wt + c_2 \sin\;wt where c_1, c_2. Therefore substituting, \Rightarrow \large\frac{d^2x}{dt^2}$$ = -w^2x \rightarrow \large\frac{d^2x}{dt^2}$$+ w^2x = 0$