Question

The solution of the differential equation \( \frac{d^2y}{dx^2} = 0 \) represents

• A. all circles in a plane
• B. all straight lines in a plane
• C. all parabolas in a plane
• D. all ellipses in a plane

Hint:

When the second derivative of a function is zero, the original function must be linear (i.e., a straight line).

Answer 1

The Correct Option is B

We are given the second-order differential equation: \[ \frac{d^2y}{dx^2} = 0 \] Step 1: Integrate once with respect to \( x \)
\[ \frac{dy}{dx} = C_1 \quad \text{(where \( C_1 \) is the constant of integration)} \] Step 2: Integrate again
\[ y = C_1x + C_2 \quad \text{(where \( C_2 \) is another constant)} \] Step 3: Analyze the result
The solution \( y = C_1x + C_2 \) is the general equation of a straight line.
Therefore, the differential equation \( \frac{d^2y}{dx^2} = 0 \) represents all straight lines in a plane.