Concept:
In differential equations, Order and Degree are fundamental properties.
- Order: The highest order derivative present in the equation.
- Degree: The power of the highest order derivative (when the equation is a polynomial in derivatives).
Step 1: Understanding the Question:
We are given the equation \(\dfrac{d^2 y}{dx^2} + \left(\dfrac{dy}{dx}\right)^3 = 0\) and need to identify its order.
Step 2: Key Formula or Approach:
Look for all the derivatives present:
- \(\dfrac{dy}{dx}\) is a first-order derivative.
- \(\dfrac{d^2y}{dx^2}\) is a second-order derivative.
Step 3: Detailed Solution:
1. The equation contains a first derivative \((y')\) and a second derivative \((y'')\).
2. The highest derivative present is the second derivative, \(\dfrac{d^2 y}{dx^2}\).
3. Therefore, the order is 2.
(Note: The power 3 belongs to the first derivative, so it affects the degree of that term, but not the order of the entire equation.)
Step 4: Final Answer:
The order of the differential equation is 2.