Differential Equation: dy/dx = (y - x)^2

Method: Separation of Variables

Step 1: Rearrange the equation
The given differential equation is already in a suitable form for separation of variables.

Step 2: Separate the variables
Write the equation as:
1/(y-x)^2 dy = dx

Step 3: Integrate both sides
Integrate both sides of the equation with respect to their respective variables.
∫(1/(y-x)^2) dy = ∫dx

Step 4: Evaluate the integrals
The integral of the left side can be evaluated using substitution. Let u = y - x, then du = dy.
∫(1/u^2) du = ∫dx
-1/u = x + C1
-1/(y-x) = x + C1

Step 5: Solve for y
To solve for y, rearrange the equation:
y - x = -1/(x + C1)
y = -1/(x + C1) + x

Step 6: Simplify the equation
Combine the terms on the right side to simplify the equation:
y = (-1 + x(x + C1))/(x + C1)

Step 7: Final Solution
The general solution to the given differential equation is:
y = (-1 + x(x + C1))/(x + C1)

Explanation:
The given differential equation is solved using the method of separation of variables. By separating the variables and integrating both sides, we obtain the integral of (1/(y-x)^2) with respect to y and the integral of 1 with respect to x. Evaluating the integrals and solving for y gives the general solution. The constant of integration, C1, is introduced during the integration process and accounts for the family of solutions to the differential equation.