Question

For the differential equation \( \frac{d^3 y}{dx^3} = 0 \), \( y = ax^2 + bx + c \) is:

• A. The general solution
• B. A particular solution
• C. Not a solution
• D. A solution, but not a particular solution

Hint:

For higher order differential equations: - Each integration introduces a constant. - Order = number of arbitrary constants. - Check form of solution against the integrated form.

Answer 1

The Correct Option is A

We are given: \[ \frac{d^3 y}{dx^3} = 0 \] Integrating thrice: \[ \frac{d^2 y}{dx^2} = A, \quad \frac{dy}{dx} = Ax + B, \quad y = \frac{A}{2}x^2 + Bx + C \] Thus the general solution is a quadratic in \( x \), which matches the form: \[ y = ax^2 + bx + c \]