If y = sin (l n x) , then which one of the following is correct ?

Options

A. `(d^(2) y)/(dx^(2)) + y = 0`
B. `(d^(2) y)/(dx^(2)) = 0 `
C. `x^(2) "" (d^(2) y)/(dx^(2)) + x "" (dy)/(dx) + y = 0`
D. `x^(2) "" (d^(2) y)/(dx^(2)) - x"" (dy)/(dx) + y = 0`

Answer

To solve the problem where \( y = \sin(\ln x) \), we need to find the correct differential equation involving \( y \). Let's go through the steps to derive the equation.

### Step 1: Differentiate \( y \)
Given:
\[ y = \sin(\ln x) \]
We will differentiate \( y \) with respect to \( x \).

Using the chain rule:
\[ \frac{dy}{dx} = \cos(\ln x) \cdot \frac{d}{dx}(\ln x) \]
Since \( \frac{d}{dx}(\ln x) = \frac{1}{x} \), we have:
\[ \frac{dy}{dx} = \cos(\ln x) \cdot \frac{1}{x} = \frac{\cos(\ln x)}{x} \]

### Step 2: Multiply by \( x \)
Now, we multiply both sides by \( x \):
\[ x \frac{dy}{dx} = \cos(\ln x) \]

### Step 3: Differentiate again
Next, we differentiate \( \frac{dy}{dx} \) again to find the second derivative \( \frac{d^2y}{dx^2} \).

Using the product rule on \( x \frac{dy}{dx} \):
\[ \frac{d}{dx}(x \frac{dy}{dx}) = \frac{d}{dx}(x) \cdot \frac{dy}{dx} + x \cdot \frac{d^2y}{dx^2} \]
This gives us:
\[ \frac{dy}{dx} + x \frac{d^2y}{dx^2} \]

### Step 4: Set up the equation
From our previous step, we know:
\[ \frac{d}{dx}(x \frac{dy}{dx}) = \frac{d}{dx}(\cos(\ln x)) \]
Using the chain rule again:
\[ \frac{d}{dx}(\cos(\ln x)) = -\sin(\ln x) \cdot \frac{d}{dx}(\ln x) = -\sin(\ln x) \cdot \frac{1}{x} \]

### Step 5: Combine results
Setting the two expressions equal:
\[ \frac{dy}{dx} + x \frac{d^2y}{dx^2} = -\frac{\sin(\ln x)}{x} \]

### Step 6: Replace \( \sin(\ln x) \) with \( y \)
Since \( y = \sin(\ln x) \), we can substitute:
\[ \frac{dy}{dx} + x \frac{d^2y}{dx^2} = -\frac{y}{x} \]

### Step 7: Rearranging the equation
Multiplying through by \( x \) gives:
\[ x^2 \frac{d^2y}{dx^2} + x \frac{dy}{dx} + y = 0 \]

### Final Result
Thus, the correct differential equation is:
\[ x^2 \frac{d^2y}{dx^2} + x \frac{dy}{dx} + y = 0 \]