Find the differential equation governing all the straight lines `(x)/(a)+(y)/(b)=1, a ne 0, b ne 0`.

Answer

To find the differential equation governing all the straight lines given by the equation \(\frac{x}{a} + \frac{y}{b} = 1\) where \(a \neq 0\) and \(b \neq 0\), we will follow these steps:

### Step 1: Differentiate the given equation
We start with the equation:
\[ \frac{x}{a} + \frac{y}{b} = 1 \]
Differentiating both sides with respect to \(x\):
\[ \frac{d}{dx}\left(\frac{x}{a}\right) + \frac{d}{dx}\left(\frac{y}{b}\right) = \frac{d}{dx}(1) \]
This gives:
\[ \frac{1}{a} + \frac{1}{b} \frac{dy}{dx} = 0 \]

### Step 2: Solve for \(\frac{dy}{dx}\)
Rearranging the equation from Step 1:
\[ \frac{1}{b} \frac{dy}{dx} = -\frac{1}{a} \]
Multiplying both sides by \(b\):
\[ \frac{dy}{dx} = -\frac{b}{a} \]

### Step 3: Differentiate again
Since there are two constants \(a\) and \(b\) in the original equation, we need to differentiate again to eliminate these constants. Differentiating \(\frac{dy}{dx} = -\frac{b}{a}\) with respect to \(x\):
\[ \frac{d^2y}{dx^2} = 0 \]
This means that the second derivative of \(y\) with respect to \(x\) is zero.

### Step 4: Write the differential equation
The result from Step 3 gives us the differential equation:
\[ \frac{d^2y}{dx^2} = 0 \]

### Final Result
Thus, the differential equation governing all the straight lines of the form \(\frac{x}{a} + \frac{y}{b} = 1\) is:
\[ \frac{d^2y}{dx^2} = 0 \]

---