Form the differential equation representing the family of curves `y=(A)/(x)+5`, by eliminating the arbitrary constant A.

Answer

To form the differential equation representing the family of curves given by the equation \( y = \frac{A}{x} + 5 \), we will eliminate the arbitrary constant \( A \) through differentiation. Here are the steps to derive the differential equation:

### Step 1: Differentiate the given equation
We start with the equation:
\[ y = \frac{A}{x} + 5 \]
To eliminate \( A \), we first differentiate both sides with respect to \( x \):
\[ \frac{dy}{dx} = -\frac{A}{x^2} \]
The derivative of \( 5 \) is \( 0 \).

### Step 2: Solve for \( A \)
From the differentiated equation, we can express \( A \) in terms of \( y \) and \( x \):
\[ A = -x^2 \frac{dy}{dx} \]

### Step 3: Substitute \( A \) back into the original equation
Now, we substitute \( A \) back into the original equation:
\[ y = \frac{-x^2 \frac{dy}{dx}}{x} + 5 \]
This simplifies to:
\[ y = -x \frac{dy}{dx} + 5 \]

### Step 4: Rearrange to form the differential equation
Rearranging the equation gives us:
\[ x \frac{dy}{dx} + y = 5 \]

### Final Differential Equation
Thus, the differential equation representing the family of curves is:
\[ x \frac{dy}{dx} + y = 5 \]

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