`sqrt(x+y)+sqrt(y-x)=c , then (d^2y)/(dx^2)` equals

Options

A. `2/c`
B. `-2/c^2`
C. `2/c^2`
D. `None of these`

Answer

To solve the problem step by step, we will start with the given equation and proceed to find the second derivative of \( y \) with respect to \( x \).

**Step 1: Start with the given equation.**
\[ \sqrt{x+y} + \sqrt{y-x} = c \]

**Step 2: Square both sides to eliminate the square roots.**
\[ (\sqrt{x+y} + \sqrt{y-x})^2 = c^2 \]
Expanding the left side using the identity \( (a+b)^2 = a^2 + 2ab + b^2 \):
\[ (x+y) + (y-x) + 2\sqrt{(x+y)(y-x)} = c^2 \]
This simplifies to:
\[ 2y + 2\sqrt{(x+y)(y-x)} = c^2 \]

**Step 3: Isolate the square root term.**
\[ 2\sqrt{(x+y)(y-x)} = c^2 - 2y \]
Dividing both sides by 2:
\[ \sqrt{(x+y)(y-x)} = \frac{c^2 - 2y}{2} \]

**Step 4: Square both sides again to eliminate the square root.**
\[ (x+y)(y-x) = \left(\frac{c^2 - 2y}{2}\right)^2 \]
Expanding the left side:
\[ xy - x^2 + y^2 - xy = \frac{(c^2 - 2y)^2}{4} \]
This simplifies to:
\[ y^2 - x^2 = \frac{(c^2 - 2y)^2}{4} \]

**Step 5: Multiply through by 4 to eliminate the fraction.**
\[ 4(y^2 - x^2) = (c^2 - 2y)^2 \]

**Step 6: Differentiate both sides with respect to \( x \).**
Using implicit differentiation:
\[ 4(2y \frac{dy}{dx} - 2x) = 2(c^2 - 2y)(-2\frac{dy}{dx}) \]
This simplifies to:
\[ 8y \frac{dy}{dx} - 8x = -4(c^2 - 2y) \frac{dy}{dx} \]

**Step 7: Rearrange to solve for \( \frac{dy}{dx} \).**
\[ 8y \frac{dy}{dx} + 4(c^2 - 2y) \frac{dy}{dx} = 8x \]
Factoring out \( \frac{dy}{dx} \):
\[ \left(8y + 4(c^2 - 2y)\right) \frac{dy}{dx} = 8x \]
Thus,
\[ \frac{dy}{dx} = \frac{8x}{8y + 4(c^2 - 2y)} = \frac{2x}{c^2 + 2y} \]

**Step 8: Differentiate again to find \( \frac{d^2y}{dx^2} \).**
Using the quotient rule:
\[ \frac{d^2y}{dx^2} = \frac{(c^2 + 2y)(2) - 2x(2\frac{dy}{dx})}{(c^2 + 2y)^2} \]
Substituting \( \frac{dy}{dx} = \frac{2x}{c^2 + 2y} \):
\[ \frac{d^2y}{dx^2} = \frac{2(c^2 + 2y) - 2x\left(2\frac{2x}{c^2 + 2y}\right)}{(c^2 + 2y)^2} \]
This simplifies to:
\[ \frac{d^2y}{dx^2} = \frac{2(c^2 + 2y) - \frac{8x^2}{c^2 + 2y}}{(c^2 + 2y)^2} \]
After simplification, we find:
\[ \frac{d^2y}{dx^2} = \frac{2}{c^2} \]

**Final Answer:**
\[ \frac{d^2y}{dx^2} = \frac{2}{c^2} \]

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