If `f' ((x)/(y)). f((y)/(x))=(x^(2)+y^(2))/(xy) AA x,y in R^(+) and f(1)=1`, then `f^(2) (x)` is

Options

A. `x+In x`
B. `x^(2)+2Inx`
C. `x^(2)+In x`
D. `x+2In x`

Explanation

To solve the problem, we need to find \( f^2(x) \) given the equation:

\[ f' \left( \frac{x}{y} \right) f \left( \frac{y}{x} \right) = \frac{x^2 + y^2}{xy} \]

with the condition \( f(1) = 1 \).

### Step 1: Interchange \( x \) and \( y \)

We start by interchanging \( x \) and \( y \) in the original equation:

\[ f' \left( \frac{y}{x} \right) f \left( \frac{x}{y} \right) = \frac{y^2 + x^2}{yx} \]

Since \( x^2 + y^2 = y^2 + x^2 \), we can equate both equations:

\[ f' \left( \frac{x}{y} \right) f \left( \frac{y}{x} \right) = f' \left( \frac{y}{x} \right) f \left( \frac{x}{y} \right) \]

### Step 2: Rearranging the Equation

From the above equality, we can rearrange it to:

\[ f' \left( \frac{y}{x} \right) \frac{1}{f \left( \frac{y}{x} \right)} = \frac{1}{f \left( \frac{x}{y} \right)} f' \left( \frac{x}{y} \right) \]

### Step 3: Integrate Both Sides

Now we can integrate both sides. Since the left side can be expressed in the form \( \frac{1}{f(u)} f'(u) \), we integrate:

\[ \int \frac{1}{f(u)} f'(u) \, du = \ln |f(u)| + C \]

This gives us:

\[ \ln \left( f \left( \frac{y}{x} \right) \right) = \ln \left( f \left( \frac{x}{y} \right) \right) + C \]

### Step 4: Using the Condition \( f(1) = 1 \)

Now, we set \( y = x \):

\[ f(1) = f(1) + C \implies C = 0 \]

Thus, we have:

\[ f \left( \frac{y}{x} \right) = f \left( \frac{x}{y} \right) \]

### Step 5: Set \( y = 1 \)

Next, we set \( y = 1 \) in the original equation:

\[ f' (x) f(1/x) = x^2 + 1/x \]

Using the fact that \( f(1) = 1 \):

\[ f' (x) f(1/x) = x + 1/x \]

### Step 6: Substitute \( f(1/x) \)

From our earlier result \( f(1/x) = f(x) \):

\[ f' (x) f(x) = x + 1/x \]

### Step 7: Rewrite and Integrate

This can be rewritten as:

\[ f(x) f'(x) = x + \frac{1}{x} \]

Now, we can integrate both sides:

\[ \int f(x) \, df = \int \left( x + \frac{1}{x} \right) \, dx \]

This gives:

\[ \frac{f^2(x)}{2} = \frac{x^2}{2} + \ln |x| + C \]

### Step 8: Solve for \( C \)

Using \( f(1) = 1 \):

\[ \frac{1^2}{2} = \frac{1^2}{2} + \ln(1) + C \implies C = 0 \]

### Step 9: Final Expression for \( f^2(x) \)

Thus, we have:

\[ \frac{f^2(x)}{2} = \frac{x^2}{2} + \ln |x| \]

Multiplying through by 2 gives:

\[ f^2(x) = x^2 + 2 \ln |x| \]

### Final Answer

Thus, the required expression for \( f^2(x) \) is:

\[ \boxed{x^2 + 2 \ln x} \]