` (x-y)(1-(dy)/(dx))=e^(x)`

Answer

To solve the differential equation \((x - y)(1 - \frac{dy}{dx}) = e^x\), we can follow these steps:

### Step 1: Substitute \(u = x - y\)
Let \(u = x - y\). Then, we can express \(y\) in terms of \(u\):
\[ y = x - u \]

### Step 2: Differentiate \(u\)
Now, differentiate \(u\) with respect to \(x\):
\[ \frac{du}{dx} = \frac{d}{dx}(x - y) = 1 - \frac{dy}{dx} \]

### Step 3: Rewrite the differential equation
Substituting \(u\) and \(\frac{du}{dx}\) into the original equation gives:
\[ u(1 - \frac{dy}{dx}) = e^x \]
This can be rewritten using our expression for \(\frac{dy}{dx}\):
\[ u \cdot \frac{du}{dx} = e^x \]

### Step 4: Separate variables
Now, we can separate the variables:
\[ u \, du = e^x \, dx \]

### Step 5: Integrate both sides
Integrate both sides:
\[ \int u \, du = \int e^x \, dx \]
This results in:
\[ \frac{u^2}{2} = e^x + C \]

### Step 6: Substitute back for \(u\)
Now, substitute back \(u = x - y\):
\[ \frac{(x - y)^2}{2} = e^x + C \]

### Step 7: Multiply through by 2
Multiply the entire equation by 2 to simplify:
\[ (x - y)^2 = 2e^x + 2C \]

### Final Solution
Thus, the solution to the differential equation is:
\[ (x - y)^2 = 2e^x + C' \]
where \(C' = 2C\) is a constant.