Higher Derivatives Question

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March 21st 2011, 01:15 AM
AeroScizor

Higher Derivatives Question

Hi all,
Here goes the question:
Given that $y=xsin3x+cos3x$ , show that $x^2\frac{d^2y}{dx^2}+2y+4x^2y=0$ .

I am quite comfortable in deriving the normal and higher derivatives(*Just to make sure I am on the right track, is $\frac{dy}{dx}$ =sin3x-3sin3x+3xcos3x?) and am more concerned about the 'showing' part. Hopefully someone can guide me along.

Another one:
Given that $xy=sinx$ , prove that $\frac{d^2y}{dx^2}+\frac{2}{x}\frac{dy}{dx}+y=0$ .

It seems like a typical implicit diff. question other than the higher derivatives part. I haven't really learn how to derive higher derivatives using implicit diff.

Any help is appreciated. Thanks in advance!
March 21st 2011, 01:19 AM
Prove It

When you have found $\displaystyle \frac{dy}{dx}$ and $\displaystyle \frac{d^2y}{dx^2}$ , substitute them and $\displaystyle y$ into the LHS of your equation. Show that it simplifies to the RHS.
March 21st 2011, 01:53 AM
AeroScizor

Quote:

Originally Posted by Prove It

When you have found $\displaystyle \frac{dy}{dx}$ and $\displaystyle \frac{d^2y}{dx^2}$ , substitute them and $\displaystyle y$ into the LHS of your equation. Show that it simplifies to the RHS.

Cheers. Your reply was short and concise but manage to set my thinking straight. Now I am proud that I am finally able to attempt the question. Thanks again.