Question

Determine whether the given relation is an implicit solution to the given differential equation. Assume that the relationship does define y implicitly as a function of x and use implicit differentiation.

y- lny= x^2+1 , dy/dx= 2xy/y-1

Answer 1

See explanation below

Explanation:

Given the expression

y- lny= x^2+1

Differentiate implicitly with respect to x

dy/dx - 1/y dy/dx = 2x+0

Factor out dy/dx

dy/dx (1 - 1/y) = 2x

dy/dx (y-1/y) =2x

Cross multiply

dy/dx (y-1) = 2xy

Divide both sides by y-1

dy/dx (y-1)/y-1 = 2xy/y-1

dy/dx = 2xy/y-1

Hence the given relation is an implicit solution to the given differential equation