Solution of the Differential Equation

The given differential equation is: (dy/dx)=[((1 x)y)/((y-1)x)]

To solve this differential equation, we can use the method of separation of variables.

Separation of Variables

We can separate the variables y and x by bringing all the y terms to one side and all the x terms to the other side:

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

Integrating both sides, we get:

∫(dy/((y-1)y)) = ∫(dx/x)

Solving the Integrals

The integral on the left side can be solved using partial fraction decomposition:

∫(dy/((y-1)y)) = ∫(A/(y-1) + B/y) dy

Multiplying both sides by (y-1)y, we get:

1 = A(y) + B(y-1)

Setting y = 0 and y = 1, we get:

A = 1 and B = -1

Substituting these values in the partial fraction decomposition, we get:

∫(dy/((y-1)y)) = ∫(1/(y-1) - 1/y) dy

Integrating both sides, we get:

ln|y-1| - ln|y| = ln|x| + C

where C is the constant of integration.

Simplifying the Expression

Using the properties of logarithms, we can simplify this expression as:

ln|(y-1)/y| = ln|x| + C

Taking the exponential of both sides, we get:

|(y-1)/y| = e^(ln|x|+C)

Simplifying further, we get:

|(y-1)/y| = Kx

where K = e^C.

Solving for y

We can solve for y by breaking the absolute value into two cases:

Case 1: (y-1)/y = Kx

Solving for y, we get:

y = 1/(1-Kx)

Case 2: (y-1)/y = -Kx

Solving for y, we get:

y = 1/(1+Kx)

Final Solution

The general solution of the differential equation is given by:

y = 1/(1-Kx) or y = 1/(1+Kx)

where K = e^C.

Comparing with Options

Comparing with the given options, we can see that the correct option is:

logxy x-y= c

where c is a constant of integration.