SOLUTION: If y=x²+y²=2y, find dy/dx in terms of x and y. Prove that d²y/dx²=1/(1-y)³

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Question 1204134: If y=x²+y²=2y, find dy/dx in terms of x and y. Prove that d²y/dx²=1/(1-y)³
Found 3 solutions by ikleyn, Edwin McCravy, math_tutor2020:
Answer by ikleyn(52790) About Me  (Show Source):
You can put this solution on YOUR website!
.

As equation is written in the post with two equation signs,

it contradicts to basic elementary Math grammar rules and is not readable.



Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
I'm sorry but this
 
y=x²+y²=2y

makes no sense unless x and y were both equal to zero.

y just doesn't equal to 2y any other time. That is not what you meant.

please tell me what you meant in the space provided below for a thank you
note, and I'll help you.

Edwin

Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Through a bit of trial-and-error, it appears the given equation should be x%5E2%2By%5E2+=+2y.
Please be more careful when entering the equation.

The 1st derivative would be: dy%2Fdx+=+x%2F%281-y%29
The scratch work is shown in the next section.

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x%5E2%2By%5E2+=+2y

expr%28d%2F%28dx%29%29%28+x%5E2+%2B+y%5E2+%29+=+expr%28d%2F%28dx%29%29%282y%29



2x+%2B+2y%2Aexpr%28%28dy%29%2F%28dx%29%29+=+2%2Aexpr%28%28dy%29%2F%28dx%29%29

2x+=+2%2Aexpr%28%28dy%29%2F%28dx%29%29+-++2y%2Aexpr%28%28dy%29%2F%28dx%29%29

2x+=+%282+-++2y%29%2Aexpr%28%28dy%29%2F%28dx%29%29

%28dy%29%2F%28dx%29+=+%282x%29%2F%282-2y%29

%28dy%29%2F%28dx%29+=+%282x%29%2F%282%281-y%29%29

%28dy%29%2F%28dx%29+=+x%2F%281-y%29
This will be used later.

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We then apply another round of implicit differentiation with respect to x.

%28dy%29%2F%28dx%29+=+x%2F%281-y%29

%28d%5E2y%29%2F%28dx%5E2%29+=+expr%28d%2F%28dx%29%29%28x%2F%281-y%29%29

Quotient Rule







From here we plug in the first derivative








%28d%5E2y%29%2F%28dx%5E2%29+=+%28%281-y%29%5E2%2Bx%5E2%29%2F%28+%281-y%29%5E3+%29

%28d%5E2y%29%2F%28dx%5E2%29+=+%28%281-2y%2By%5E2%29%2Bx%5E2%29%2F%28+%281-y%29%5E3+%29

%28d%5E2y%29%2F%28dx%5E2%29+=+%281-2y%2B%28x%5E2%2By%5E2%29%29%2F%28+%281-y%29%5E3+%29



%28d%5E2y%29%2F%28dx%5E2%29+=+%281-2y%2Bhighlight%282y%29%29%2F%28+%281-y%29%5E3+%29 Use the equation x^2+y^2 = 2y

%28d%5E2y%29%2F%28dx%5E2%29+=+1%2F%28+%281-y%29%5E3+%29


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Here's another way to do it

%28dy%29%2F%28dx%29+=+x%2F%281-y%29

expr%28%28dy%29%2F%28dx%29%29%2A%281-y%29+=+x

Product Rule



expr%28%28d%5E2y%29%2F%28dx%5E2%29%29%2A%281-y%29+-+%28%28dy%29%2F%28dx%29%29%5E2+=+1

expr%28%28d%5E2y%29%2F%28dx%5E2%29%29%2A%281-y%29+-+%28x%2F%281-y%29%29%5E2+=+1 Plug in dy/dx = x/(1-y)

expr%28%28d%5E2y%29%2F%28dx%5E2%29%29%2A%281-y%29+=+1+%2B+%28x%2F%281-y%29%29%5E2







%28d%5E2y%29%2F%28dx%5E2%29+=+%28%281-2y%2By%5E2%29%2Bx%5E2%29%2F%28%281-y%29%5E3%29%29

%28d%5E2y%29%2F%28dx%5E2%29+=+%281-2y%2B%28x%5E2%2By%5E2%29%29%2F%28%281-y%29%5E3%29%29

%28d%5E2y%29%2F%28dx%5E2%29+=+%281-2y%2B2y%29%2F%28%281-y%29%5E3%29%29 Replace x^2+y^2 with 2y

%28d%5E2y%29%2F%28dx%5E2%29+=+1%2F%28%281-y%29%5E3%29%29