SOLUTION: find an expression for d^2y/dx^2 if 2x^2 =3y^2=9. Write your answer in terms of y( no x's) the answer should be in simplest form.

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Question 389723: find an expression for d^2y/dx^2 if 2x^2 =3y^2=9. Write your answer in terms of y( no x's) the answer should be in simplest form.
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
find an expression for %28d%5E2y%29%2F%28dx%5E2%29 if 2x%5E2+=3y%5E2%2B9.
Write your answer in terms of y( no x's) the answer
should be in simplest form.
2x%5E2+=3y%5E2%2B9 
4x+=6y%2Aexpr%28dy%2Fdx%29%2B0
4x+=6y%2Aexpr%28dy%2Fdx%29
2x+=3y%2Aexpr%28dy%2Fdx%29
%282x%29%2F%283y%29=expr%28dy%2Fdx%29
dy%2Fdx=%282x%29%2F%283y%29
%28d%5E2y%29%2F%28dx%5E2%29=%283y%2A2-2x%2A3expr%28dy%2Fdx%29%29%2F%28%283y%29%5E2%29

Simplify, substituting %282x%29%2F%283y%29 for %28dy%29%2F%28dx%29

%28d%5E2y%29%2F%28dx%5E2%29=%286y-2x%2A3%2Aexpr%28%282x%29%2F%283y%29%29%29%2F%289y%5E2%29

%28d%5E2y%29%2F%28dx%5E2%29=%286y-4x%5E2%29%2Fy

Solve the original equation for x%5E2

2x%5E2+=3y%5E2%2B9

x%5E2+=%283y%5E2%2B9%29%2F2

Substitute that in %28d%5E2y%29%2F%28dx%5E2%29=%286y-4x%5E2%29%2Fy

%28d%5E2y%29%2F%28dx%5E2%29=%286y-4%2A%28%283y%5E2%2B9%29%2F2%29%29%2Fy

Cancel the 2 into the 4:

%28d%5E2y%29%2F%28dx%5E2%29=%286y-2%2A%283y%5E2%2B9%29%29%2Fy

%28d%5E2y%29%2F%28dx%5E2%29=%286y-6y%5E2-18%29%2Fy

There are several different ways to rewrite that if you like:

%28d%5E2y%29%2F%28dx%5E2%29=6%28y-y%5E2-3%29%2Fy

or

%28d%5E2y%29%2F%28dx%5E2%29=6-6y-18%2Fy

or 

%28d%5E2y%29%2F%28dx%5E2%29=expr%28-6%2Fy%29%2A%28y%5E2-y%2B3%29
 
Edwin