Ask Questions, Get Answers

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.


Note: Clay6 has edited the question for clarity.

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$
Help Clay6 to be free
Clay6 needs your help to survive. We have roughly 7 lakh students visiting us monthly. We want to keep our services free and improve with prompt help and advanced solutions by adding more teachers and infrastructure.

A small donation from you will help us reach that goal faster. Talk to your parents, teachers and school and spread the word about clay6. You can pay online or send a cheque.

Thanks for your support.
Please choose your payment mode to continue
Home Ask Homework Questions
Your payment for is successful.