If `sqrt(x+y)+sqrt(y-x)=2`, then the value of `(d^(2)y)/(dx^(2))` is equal to

Answer

To solve the problem step by step, we start with the given equation:

\[ \sqrt{x+y} + \sqrt{y-x} = 2 \]

### Step 1: Square both sides
We square both sides to eliminate the square roots.

\[ (\sqrt{x+y} + \sqrt{y-x})^2 = 2^2 \]

Expanding the left side using the formula \((a+b)^2 = a^2 + b^2 + 2ab\):

\[ (x+y) + (y-x) + 2\sqrt{(x+y)(y-x)} = 4 \]

### Step 2: Simplify the equation
Now, simplify the left side:

\[ y + y + 2\sqrt{(x+y)(y-x)} = 4 \]

This simplifies to:

\[ 2y + 2\sqrt{(x+y)(y-x)} = 4 \]

### Step 3: Isolate the square root
Next, isolate the square root term:

\[ 2\sqrt{(x+y)(y-x)} = 4 - 2y \]

Dividing both sides by 2:

\[ \sqrt{(x+y)(y-x)} = 2 - y \]

### Step 4: Square both sides again
Square both sides again to eliminate the square root:

\[ (x+y)(y-x) = (2-y)^2 \]

### Step 5: Expand both sides
Now, expand both sides:

Left side:

\[ xy - x^2 + y^2 - xy = y^2 - x^2 \]

Right side:

\[ (2-y)(2-y) = 4 - 4y + y^2 \]

### Step 6: Set the equation
Setting both sides equal gives:

\[ y^2 - x^2 = 4 - 4y + y^2 \]

### Step 7: Simplify
Subtract \(y^2\) from both sides:

\[ -x^2 = 4 - 4y \]

Rearranging gives:

\[ x^2 = 4y - 4 \]

### Step 8: Differentiate both sides
Now, differentiate both sides with respect to \(x\):

\[ 2x \frac{dx}{dx} = 4 \frac{dy}{dx} \]

This simplifies to:

\[ 2x = 4 \frac{dy}{dx} \]

### Step 9: Solve for \(\frac{dy}{dx}\)
Dividing both sides by 4:

\[ \frac{dy}{dx} = \frac{x}{2} \]

### Step 10: Differentiate again to find \(\frac{d^2y}{dx^2}\)
Now, differentiate \(\frac{dy}{dx}\) again:

\[ \frac{d^2y}{dx^2} = \frac{1}{2} \frac{dx}{dx} = \frac{1}{2} \]

Thus, the final answer is:

\[ \frac{d^2y}{dx^2} = \frac{1}{2} \]