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Questions  >>  CBSE XII  >>  Math  >>  Differential Equations

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

From the user:

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.


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1 Answer

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$
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