Questions: Find the particular solution of the following differential equation, given that the curve passes through the point (3,2). Give your answer in its simplest form: 12/x dy/dx = 6/y

Find the particular solution of the differential equation given the initial condition.

Separate the variables.

Given the differential equation:
\[ \frac{12}{x} \frac{d y}{d x}=\frac{6}{y} \]
Rearrange to separate variables:
\[ 12y \, dy = 6x \, dx \]
Simplify:
\[ 2y \, dy = x \, dx \]

Integrate both sides.

Integrate the separated equation:
\[ \int 2y \, dy = \int x \, dx \]
This results in:
\[ y^2 = \frac{x^2}{2} + C \]

Solve for \(y\).

Isolate \(y\):
\[ y = \pm \sqrt{\frac{x^2}{2} + C} \]

Apply the initial condition.

Given the point \((3, 2)\), substitute \(x = 3\) and \(y = 2\):
\[ 2 = \pm \sqrt{\frac{3^2}{2} + C} \]
\[ 2 = \pm \sqrt{\frac{9}{2} + C} \]
Square both sides:
\[ 4 = \frac{9}{2} + C \]
Solve for \(C\):
\[ C = 4 - \frac{9}{2} = \frac{8}{2} - \frac{9}{2} = -\frac{1}{2} \]
Since \(y=2\) is positive, take the positive square root.

Write the particular solution.

Substitute \(C = -\frac{1}{2}\) back into the equation:
\[ y = \sqrt{\frac{x^2}{2} - \frac{1}{2}} \]
Simplify:
\[ y = \sqrt{\frac{x^2 - 1}{2}} \]

\(\boxed{y = \sqrt{\frac{x^2 - 1}{2}}}\)

\(\boxed{y = \sqrt{\frac{x^2 - 1}{2}}}\)