If #y^2=2xy+2x# prove that #(x-y)(d^2y)/dx^2+2dy/dx-(dy/dx)^2=0# ?
1 Answer
Apr 15, 2018
We have an implicit equation:
# y^2 = 2xy + 2x #
Differentiating Implicitly, and applying the product rule we have:
# d/dx(y^2) = (2x)(d/dx y) + (d/dx 2x)(y) + d/dx(2x) #
# :. 2y dy/dx = (2x)(dy/dx) + (2)(y) + 2 #
# :. (x-y)dy/dx + y + 1 =0 #
Differentiating this result, again using implicit differentiation and the product rule, we get
# (x-y)(d/dx dy/dx) + (d/dx(x-y))(dy/dx) + (d/dx y) + d/dx(1) =0 #
# :. (x-y)((d^2y)/(dx^2)) + (1-dy/dx)dy/dx + dy/dx = 0 #
# :. (x-y)(d^2y)/(dx^2) + dy/dx - (dy/dx)^2 + dy/dx = 0 #
# :. (x-y)(d^2y)/(dx^2) + 2dy/dx - (dy/dx)^2 = 0 \ \ \ # QED
