Question

What is the order of the differential equation \( \dfrac{d^2 y}{dx^2} + \left(\dfrac{dy}{dx}\right)^3 = 0 \)?

• A. \(1\)
• B. \(2\)
• C. \(3\)
• D. \(4\)

Hint:

Order of a differential equation = highest derivative present in the equation. The exponent of the derivative does not affect the order.

Answer 1

The Correct Option is B

Concept: The order of a differential equation is defined as the highest order derivative present in the equation. In general:
• First derivative \( \frac{dy}{dx} \) → first order
• Second derivative \( \frac{d^2y}{dx^2} \) → second order
• Third derivative \( \frac{d^3y}{dx^3} \) → third order The order depends only on the highest derivative and not on its power.

Step 1: Identify all derivatives present in the equation. Given differential equation: \[ \dfrac{d^2 y}{dx^2} + \left(\dfrac{dy}{dx}\right)^3 = 0 \] Here we observe two derivatives:
• First derivative: \( \dfrac{dy}{dx} \)
• Second derivative: \( \dfrac{d^2y}{dx^2} \)

Step 2: Determine the highest order derivative. The highest order derivative appearing in the equation is: \[ \dfrac{d^2y}{dx^2} \] This is a second order derivative.

Step 3: Note about powers of derivatives. Even though \( \left(\dfrac{dy}{dx}\right)^3 \) contains a power of 3, the order is determined only by the derivative itself, not by its exponent. Therefore, the order of the differential equation is \[ \boxed{2} \]