Given:
We are given a differential equation: x dy/dx = (x² - x - 1)y(x - 1)

To Find:
We need to find the value of the constant k.

Approach:
To solve the differential equation, we will use the method of separation of variables. Let's go step by step to find the value of k.

Step 1: Rewrite the Differential Equation
Rewrite the given differential equation in a more convenient form:
(x² - x - 1)y(x - 1)dx - xdy = 0

Step 2: Separate the Variables
Separate the variables by moving the terms involving x to one side and the terms involving y to the other side:
(x² - x - 1)dx - xdy = (x - 1)ydx

Step 3: Integrate both sides
Integrate both sides of the equation with respect to their respective variables:
∫ (x² - x - 1)dx - ∫ xdy = ∫ (x - 1)ydx

Step 4: Evaluate the Integrals
Evaluate the integrals on both sides of the equation:
(x³/3 - x²/2 - x) - (xy - y/2) = (x²/2 - x) + C

Step 5: Simplify the Equation
Simplify the equation by combining like terms:
x³/3 - x²/2 - x - xy + y/2 = x²/2 - x + C

Step 6: Rearrange the Equation
Rearrange the equation to isolate the y term on one side:
x³/3 - x²/2 - x + x²/2 - x + xy - y/2 = C

Step 7: Simplify Further
Simplify the equation by combining like terms:
x³/3 - x + xy - y/2 = C

Step 8: Rewrite the Equation
Rewrite the equation in the form of the given equation:
(x³/3 - x + xy - y/2) - k = 0

Step 9: Compare Coefficients
Compare the coefficients of similar terms in the two equations. We can see that the coefficient of (x - 1) term on both sides should be equal:
-1/2 = -k

Step 10: Solve for k
Solve for k by multiplying both sides by -1:
k = 1/2

Final Answer:
The value of the constant k is 1/2.