Form the differential equation of the family of lines making equal intercepts on the co-ordinate axes.

Answer

To form the differential equation of the family of lines making equal intercepts on the coordinate axes, we can follow these steps:

### Step 1: Write the equation of the line with equal intercepts
The general equation of a line making equal intercepts on the coordinate axes can be expressed as:
\[ \frac{x}{a} + \frac{y}{a} = 1 \]
where \(a\) is the intercept on both the x-axis and y-axis.

### Step 2: Simplify the equation
We can simplify the equation:
\[ \frac{x + y}{a} = 1 \]
Multiplying through by \(a\) gives:
\[ x + y = a \]

### Step 3: Differentiate the equation
Now, we need to eliminate the parameter \(a\) to form the differential equation. We can differentiate both sides with respect to \(x\):
\[ \frac{d}{dx}(x + y) = \frac{d}{dx}(a) \]
This gives:
\[ 1 + \frac{dy}{dx} = 0 \]

### Step 4: Solve for \(\frac{dy}{dx}\)
Rearranging the equation, we find:
\[ \frac{dy}{dx} = -1 \]

### Step 5: Form the differential equation
The equation \(\frac{dy}{dx} = -1\) represents the differential equation of the family of lines making equal intercepts on the coordinate axes.

### Final Result
Thus, the required differential equation is:
\[ \frac{dy}{dx} + 1 = 0 \]

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