Find the particular solution that satisfies the intial condition:

xdy - (2xe^(-y/x)+y)dx = 0

To find the particular solution that satisfies the initial condition, we need to solve the given differential equation and substitute the initial condition into the solution.

Let's start by rearranging the differential equation:

xdy = (2xe^(-y/x) + y)dx
dy/y + 2e^(-y/x)/(x)dx + dx/x = 0

Now, we separate the variables by moving all the terms containing y to the left-hand side and the terms containing x to the right-hand side:

dy/y + 2e^(-y/x)/(x)dx = -dx/x

Next, we integrate both sides of the equation:

∫(dy/y) + 2∫(e^(-y/x)/(x))dx = -∫(dx/x)

Integrating the left-hand side:

ln|y| + 2∫(e^(-y/x)/(x))dx = -ln|x| + C1

We can simplify the integral of (e^(-y/x)/(x))dx:

2∫(e^(-y/x)/(x))dx = 2e^(-y/x) + C2

Now, the equation becomes:

ln|y| + 2e^(-y/x) + C2 = -ln|x| + C1

Combining the constants:

ln|y| + 2e^(-y/x) + C = -ln|x|

Where C = C1 + C2

Exponentiating both sides:

|y|e^(2e^(-y/x))e^C = 1/|x|

|y|e^(2e^(-y/x)) = k/x (where k = e^C)

Now, we substitute the initial condition into the equation:

When x = x0 and y = y0, we have:

|y0|e^(2e^(-y0/x0)) = k/x0

Solving for k:

k = x0|y0|e^(2e^(-y0/x0))

Finally, we can write the particular solution that satisfies the given initial condition as:

|y|e^(2e^(-y/x)) = (x0|y0|e^(2e^(-y0/x0)))/x

Please note that finding an explicit expression for y in terms of x may not be feasible due to the complexity of the equation. However, the equation |y|e^(2e^(-y/x)) = (x0|y0|e^(2e^(-y0/x0)))/x represents the particular solution that satisfies the initial condition.