Find #(d^2y)/(dx^2)∣_[(x,y)=(2,1)]# if y is a differentiable function of #x# satisfying the equation #x^3+2y^3 = 5xy#? Plot it?
1 Answer
May 29, 2016
Explanation:
Taking the derivative of
#3x^2+6y^2dy/dx=5y+5xdy/dx#
Rearranging and solving for
#dy/dx=(5y-3x^2)/(6y^2-5x)#
Note that
#(dy)/(dx)∣_[(x,y)=(2,1)]=(5-12)/(6-10)=7/4#
Differentiating once more:
#(d^2y)/dx^2=((5ydy/dx-6x)(6y^2-5x)-(12ydy/dx-5)(5y-3x^2))/(6y^2-5x)^2#
When evaluating this at
#(d^2y)/dx^2|_[(x,y)=(2,1)]=((5(7/4)-12)(6-10)-(12(7/4)-5)(5-12))/(6-10)^2#
#(d^2y)/dx^2|_[(x,y)=(2,1)]=((-13/4)(-4)-16(-7))/16#
#(d^2y)/dx^2|_[(x,y)=(2,1)]=(13+112)/16=125/16#