Step 1: Rewrite the given differential equation as:

$$y e^{\frac{x}{y}} dx - x e^{\frac{x}{y}} dy = y^2 dy$$

Step 2: Let $v = \frac{x}{y}$, so $x = vy$. Differentiating:

$$dx = v \, dy + y \, dv$$

Step 3: Substitute $x = vy$ and $dx = v \, dy + y \, dv$ in the equation:

$$y e^v (v \, dy + y \, dv) = \left(vy e^v + y^2\right) dy$$

Step 4: Simplify:

$$vy e^v dy + y^2 e^v dv = vy e^v dy + y^2 dy$$

The term $vy e^v dy$ cancels on both sides.

Step 5: We get:

$$y^2 e^v dv = y^2 dy$$

which simplifies to:

$$e^v dv = dy$$

(since $y \neq 0$).

Step 6: Integrate both sides:

$$\int e^v dv = \int dy$$

giving:

$$e^v = y + C$$

Step 7: Substitute back $v = \frac{x}{y}$ to get the general solution:

$$e^{\frac{x}{y}} = y + C$$