SOLVE THE QUESTION

Question

X^ZY^W=1           PROVE THAT :d²Y/dX²=Z(Z+W)/W²X²

Answer 1

Find the derivative of y with respect to x implicitly from the equation.

 

zx^{(z - 1)}y^w + x^zwy^{(w - 1)}y' = 0

 

 

Isolate y'. This means dy/dx.

 

y' = [-zx^{(z - 1)}y^w] / [x^zwy^{(w - 1)}]

 

y' = [-zx^zy^w / x] / [x^zwy^w / y]

 

dy/dx = [-zx^zy^{(w + 1)}] / [x^{(z + 1)}wy^w]

 

 

Now just take the derivative of dy/dx.  That will be d²y/dx².  Then check to see if it matches the second derivative value.