Solve the following differential equation: `xy(dy)/(dx)-y^2=(x+y)^2e^(-y//x)`

Answer

To solve the differential equation \( xy \frac{dy}{dx} - y^2 = (x+y)^2 e^{-\frac{y}{x}} \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ xy \frac{dy}{dx} - y^2 = (x+y)^2 e^{-\frac{y}{x}} \]

### Step 2: Divide by \( xy \)
Since the equation is homogeneous, we divide the entire equation by \( xy \):
\[ \frac{dy}{dx} - \frac{y}{x} = \frac{(x+y)^2}{xy} e^{-\frac{y}{x}} \]

### Step 3: Simplify the right-hand side
Now, we simplify the right-hand side:
\[ \frac{(x+y)^2}{xy} = \frac{x^2 + 2xy + y^2}{xy} = \frac{x}{y} + 2 + \frac{y}{x} \]
Thus, we rewrite the equation as:
\[ \frac{dy}{dx} - \frac{y}{x} = \left(\frac{x}{y} + 2 + \frac{y}{x}\right) e^{-\frac{y}{x}} \]

### Step 4: Substitute \( t = \frac{y}{x} \)
Let \( t = \frac{y}{x} \), then \( y = tx \) and differentiating gives:
\[ \frac{dy}{dx} = t + x \frac{dt}{dx} \]
Substituting into the equation gives:
\[ t + x \frac{dt}{dx} - t = \left(1 + 2 + t\right) e^{-t} \]
This simplifies to:
\[ x \frac{dt}{dx} = (3 + t) e^{-t} \]

### Step 5: Rearrange the equation
Rearranging gives:
\[ \frac{dt}{3 + t} = \frac{e^{-t}}{x} dx \]

### Step 6: Integrate both sides
Now we can integrate both sides:
\[ \int \frac{dt}{3 + t} = \int \frac{e^{-t}}{x} dx \]
The left side integrates to:
\[ \ln |3 + t| = -e^{-t} + C \]

### Step 7: Substitute back \( t = \frac{y}{x} \)
Substituting back \( t = \frac{y}{x} \):
\[ \ln \left|3 + \frac{y}{x}\right| = -e^{-\frac{y}{x}} + C \]

### Step 8: Final rearrangement
Exponentiating both sides gives:
\[ 3 + \frac{y}{x} = e^{-e^{-\frac{y}{x}} + C} \]
This can be rearranged to find \( y \) in terms of \( x \).

### Final Answer
The final implicit solution of the differential equation is:
\[ e^{-\frac{y}{x}}(3 + \frac{y}{x}) = C \]