Form the differential equation representing the family of curves `ax+by=0`, when 'a' and 'b' are arbitrary constants.

Answer

To form the differential equation representing the family of curves given by the equation \( ax + by = 0 \), where \( a \) and \( b \) are arbitrary constants, we can follow these steps:

### Step 1: Start with the given equation
The equation of the family of curves is:
\[ ax + by = 0 \]

### Step 2: Differentiate with respect to \( x \)
We differentiate both sides of the equation with respect to \( x \):
\[ \frac{d}{dx}(ax) + \frac{d}{dx}(by) = \frac{d}{dx}(0) \]
This gives us:
\[ a + b \frac{dy}{dx} = 0 \]

### Step 3: Solve for \( b \)
From the equation \( a + b \frac{dy}{dx} = 0 \), we can express \( b \) in terms of \( a \) and \( \frac{dy}{dx} \):
\[ b = -\frac{a}{\frac{dy}{dx}} \]

### Step 4: Substitute \( b \) back into the original equation
Now, we substitute \( b \) back into the original equation \( ax + by = 0 \):
\[ ax - \frac{a}{\frac{dy}{dx}}y = 0 \]
Factoring out \( a \) (assuming \( a \neq 0 \)):
\[ a \left( x - \frac{y}{\frac{dy}{dx}} \right) = 0 \]

### Step 5: Eliminate the constant \( a \)
Since \( a \) is an arbitrary constant, we can set \( a = 1 \) for simplicity:
\[ x - \frac{y}{\frac{dy}{dx}} = 0 \]

### Step 6: Rearranging the equation
Rearranging gives:
\[ x = \frac{y}{\frac{dy}{dx}} \]
Multiplying both sides by \( \frac{dy}{dx} \):
\[ x \frac{dy}{dx} = y \]

### Step 7: Differentiate again
Now, we differentiate \( x \frac{dy}{dx} = y \) with respect to \( x \):
\[ \frac{d}{dx}(x \frac{dy}{dx}) = \frac{dy}{dx} \]
Using the product rule on the left side:
\[ \frac{dy}{dx} + x \frac{d^2y}{dx^2} = \frac{dy}{dx} \]
This simplifies to:
\[ x \frac{d^2y}{dx^2} = 0 \]

### Step 8: Final form of the differential equation
The final form of the differential equation is:
\[ \frac{d^2y}{dx^2} = 0 \]