To find the solution of the given partial differential equation (PDE), we need to use the transformation u = w/y. Let's go step by step to understand the process.

1. Transformation:
- Given PDE: xux = u * yuy
- Let's substitute u = w/y in the PDE:
x * (w/y) * x = (w/y) * y * y * (dw/dy)
- Simplifying the equation:
x^2 * (dw/dy) = w * y

2. Separation of Variables:
- We can see that the transformed PDE involves both w and y variables. To solve it, we can separate the variables by assuming w = f(x, y).
- Let's substitute w = f(x, y) in the transformed PDE:
x^2 * (df/dy) = f * y

3. Solving the Separated ODE:
- Now, we have a separated ordinary differential equation (ODE) involving f and y.
- Dividing both sides by f * y, we get:
(1/f) * (df/dy) = 1/(x^2) * (1/y)
- Integrating both sides with respect to y:
∫(1/f) * (df/dy) dy = ∫(1/(x^2) * (1/y)) dy
- ln|f| = -1/x^2 * ln|y| + C
(where C is the constant of integration)

4. Finding the Solution:
- Exponentiating both sides, we get:
|f| = e^(-1/x^2 * ln|y| + C)
- Using the properties of logarithms, we can simplify further:
|f| = e^(ln|y| / x^2) * e^C
|f| = (|y| / x^2) * e^C

- Since f(x, y) can be positive or negative, we can remove the absolute value signs and write the solution as:
f(x, y) = ± (y / x^2) * e^C

- Finally, substituting w = f(x, y), we get the solution in terms of w:
w = ± (y / x^2) * e^C

Therefore, the transformed equation has a solution of the form w = f(x, y), which is option B.