If `y=(x+sqrt(1+x^2))^n` then `(1+x^2)(d^2y)/(dx^2)+x(dy)/(dx)`

Options

A. `n ^(2)y`
B. `y ^(-n^(2))`
C. `-y`
D. `2x ^(2)y`

Answer

To solve the problem, we need to find the expression \((1+x^2) \frac{d^2y}{dx^2} + x \frac{dy}{dx}\) given that \(y = (x + \sqrt{1+x^2})^n\).

### Step 1: Differentiate \(y\) with respect to \(x\)

Given:
\[ y = (x + \sqrt{1+x^2})^n \]

Using the chain rule:
\[ \frac{dy}{dx} = n (x + \sqrt{1+x^2})^{n-1} \left(1 + \frac{1}{2\sqrt{1+x^2}} \cdot 2x\right) \]
This simplifies to:
\[ \frac{dy}{dx} = n (x + \sqrt{1+x^2})^{n-1} \left(1 + \frac{x}{\sqrt{1+x^2}}\right) \]
Now, simplifying the term inside the brackets:
\[ 1 + \frac{x}{\sqrt{1+x^2}} = \frac{\sqrt{1+x^2} + x}{\sqrt{1+x^2}} = \frac{(x + \sqrt{1+x^2})}{\sqrt{1+x^2}} \]
Thus, we have:
\[ \frac{dy}{dx} = n (x + \sqrt{1+x^2})^{n-1} \cdot \frac{(x + \sqrt{1+x^2})}{\sqrt{1+x^2}} = n \frac{(x + \sqrt{1+x^2})^n}{\sqrt{1+x^2}} \]

### Step 2: Square both sides and differentiate again

Now, squaring both sides:
\[ (1+x^2) \left(\frac{dy}{dx}\right)^2 = n^2 (x + \sqrt{1+x^2})^{2n} \]

### Step 3: Differentiate again using the product rule

Using the product rule:
\[ \frac{d}{dx}\left((1+x^2) \left(\frac{dy}{dx}\right)^2\right) = \frac{d}{dx}(n^2 (x + \sqrt{1+x^2})^{2n}) \]

Applying the product rule on the left-hand side:
\[ \frac{d}{dx}(1+x^2) \cdot \left(\frac{dy}{dx}\right)^2 + (1+x^2) \cdot 2\frac{dy}{dx}\frac{d^2y}{dx^2} \]
This becomes:
\[ 2x \left(\frac{dy}{dx}\right)^2 + (1+x^2) \cdot 2\frac{dy}{dx}\frac{d^2y}{dx^2} \]

### Step 4: Differentiate the right-hand side

For the right-hand side:
\[ \frac{d}{dx}(n^2 (x + \sqrt{1+x^2})^{2n}) = n^2 \cdot 2n (x + \sqrt{1+x^2})^{2n-1} \cdot \left(1 + \frac{x}{\sqrt{1+x^2}}\right) \]

### Step 5: Set the equations equal

Equating both sides:
\[ 2x \left(\frac{dy}{dx}\right)^2 + (1+x^2) \cdot 2\frac{dy}{dx}\frac{d^2y}{dx^2} = n^2 \cdot 2n (x + \sqrt{1+x^2})^{2n-1} \cdot \left(1 + \frac{x}{\sqrt{1+x^2}}\right) \]

### Step 6: Solve for \((1+x^2) \frac{d^2y}{dx^2} + x \frac{dy}{dx}\)

Rearranging gives:
\[ (1+x^2) \frac{d^2y}{dx^2} + x \frac{dy}{dx} = n^2 (x + \sqrt{1+x^2})^n \]

Thus, we find:
\[ (1+x^2) \frac{d^2y}{dx^2} + x \frac{dy}{dx} = n^2 y \]

### Final Answer
\[ (1+x^2) \frac{d^2y}{dx^2} + x \frac{dy}{dx} = n^2 y \]