Differentiate the equation y+ xy = 2x+y^2 and find dy/dx
Given the equation:
y + xy = 2x + y^2
We will use implicit differentiation to find dy/dx
We will differentiate with respect to x.
==> y' + ( x'*y + x*y') = 2 + 2yy'
==> y' + y + xy' = 2+ 2yy'
Now we will group all terms with y' on one side.
==> y' + xy' -2yy' = 2 -y
Now we will factor y'.
==> y'(1 + x -2y) = (2-y)
Now we will divide by (1+x-2y)
==> y' = (2-y)/(1+x-2y)
Then the values of dy/dx = (2-y)/(1+x-2y)
We have y + xy = 2x + y^2. We have to find dy/dx
Using implicit differentiation.
dy/dx + x*dy/dx + y = 2 + 2y*dy/dx
=> dy/dx( 1 + x - 2y) = 2 - y
=> dy/dx = (2 - y) / (1 + x - 2y)
The required value of dy/dx for the given expression is dy/dx = (2 - y) / (1 + x - 2y)