Expert: Ahmed Salami - 11/10/2011

Question
Hai sir, good day.
I tried doing this question, but i can't solve it. Can you please guide me how to solve it. Thanks a lot.

8)Find y ' and y '' if x 2+4xy − y 2 = 8 . Your answer may involve x and y , but not y ' .


Thanks again.

Answer
Hi Abirami,
x² + 4xy - y² = 8
Differentiating with respect to x
2x + 4(xy' + y) - 2yy' = 0
2x + 4xy' + 4y - 2yy' = 0
2x + 4y = 2yy' - 4xy'
x + 2y = yy' - 2xy'
y' = (x + 2y)/(y - 2x)

To find y'', we go back and proceed from 2x + 4xy' + 4y - 2yy' = 0 and again differentiate with respect to x
2 + 4(xy'' + y') + 4y' - 2(yy'' + y'y') = 0
1 + 2(xy'' + y') + 2y' - (yy'' + y'y') = 0
1 + 2xy'' + 2y' + 2y' - yy'' + (y')² = 0
1 + 4y' + (y')² = yy'' - 2xy''
y'' = [1 + 4y' + (y')²]/(y - 2x)

And now you need to complete it by substituting for y' the previous result. You can get back to me.

Regards