Form the differential equation representing the family of curves `y= A sin x`, by eliminating the arbitrary constant A.

Answer

To form the differential equation representing the family of curves given by \( y = A \sin x \) by eliminating the arbitrary constant \( A \), we can follow these steps:

### Step 1: Differentiate the equation
Start with the equation:
\[ y = A \sin x \]
Differentiate both sides with respect to \( x \):
\[ \frac{dy}{dx} = A \cos x \]

### Step 2: Differentiate again
Now, differentiate \( \frac{dy}{dx} \) with respect to \( x \):
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}(A \cos x) \]
Using the product rule, we get:
\[ \frac{d^2y}{dx^2} = -A \sin x \]

### Step 3: Substitute \( A \) in terms of \( y \)
From the original equation \( y = A \sin x \), we can express \( A \) as:
\[ A = \frac{y}{\sin x} \]
Now substitute this expression for \( A \) into the second derivative equation:
\[ \frac{d^2y}{dx^2} = -\left(\frac{y}{\sin x}\right) \sin x \]
This simplifies to:
\[ \frac{d^2y}{dx^2} = -y \]

### Step 4: Rearranging the equation
Rearranging gives us the final form of the differential equation:
\[ \frac{d^2y}{dx^2} + y = 0 \]

### Final Result
Thus, the differential equation representing the family of curves \( y = A \sin x \) is:
\[ \frac{d^2y}{dx^2} + y = 0 \]

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