IF ` x ( dy )/(dx) + y=x . ( f ( x . Y) )/( f'(x.y)) ` then f ( x . Y) is equal to ( K being an arbitary constant )

Options

A. `Ke ^(x^2 //2)`
B. `Ke ^(y^2//2)`
C. `Ke^(xy//2)`
D. none of these

Answer

To solve the differential equation given by

\[ x \frac{dy}{dx} + y = x \frac{f(xy)}{f'(xy)}, \]

we will follow these steps:

### Step 1: Rearranging the Equation
We can rewrite the equation as:

\[ x \frac{dy}{dx} + y = x \frac{f(xy)}{f'(xy)}. \]

### Step 2: Recognizing the Form
Notice that we can express the left-hand side in a more convenient form. We can write:

\[ \frac{d(xy)}{dx} = x \frac{dy}{dx} + y. \]

Thus, we can rewrite the equation as:

\[ \frac{d(xy)}{dx} = \frac{f(xy)}{f'(xy)}. \]

### Step 3: Separation of Variables
Now we can separate the variables:

\[ \frac{f'(xy)}{f(xy)} d(xy) = x dx. \]

### Step 4: Integrating Both Sides
Next, we integrate both sides. Let \( t = xy \), then \( dt = f'(xy) \) and we have:

\[ \int \frac{f'(t)}{f(t)} dt = \int x dx. \]

The left-hand side integrates to:

\[ \ln |f(t)|, \]

and the right-hand side integrates to:

\[ \frac{x^2}{2} + C, \]

where \( C \) is a constant of integration.

### Step 5: Exponentiating Both Sides
Now, exponentiating both sides gives us:

\[ f(xy) = e^{\frac{x^2}{2} + C} = e^{C} e^{\frac{x^2}{2}}. \]

Let \( K = e^{C} \), which is an arbitrary constant. Thus, we have:

\[ f(xy) = K e^{\frac{x^2}{2}}. \]

### Final Result
Therefore, the solution to the original differential equation is:

\[ f(xy) = K e^{\frac{x^2}{2}}. \]

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