Find d2y/dx2 implicitly in terms of x and y.

Question

Find d²y/dx² implicitly in terms of x and y.

x^2y − 2x = 8

Please find the answer.

Answer 1

for dy/dx=y'

note if g(x,y)=x^2y, then x and y need to be positive or zero

2y*lnx=ln(2x+8)

2y'*lnx+2y/x=2/(2x+8)=2/x^2y

y'=[1/(x^2y*lnx)-y/(x*lnx]

y'' is harder

fyi

y''=(1/(ln(x))²)*[(-2x^-2y)(1+1/x)+(y/x²)(ln(x)+2)]

unless there is a different answer where y'' is function of x,y

Answer 2

This is too difficult to type here, so I wrote out the solution and compiled a PDF for you! Here's a Google Drive Link: https://drive.google.com/file/d/13S5sSIdNq7_6bQVn_zcCnTYlElUdV1ib/view?usp=sharing