Question

Find the order and degree of the differential equation \(\left(\dfrac{d^2y}{dx^2}\right)^3 + \left(\dfrac{dy}{dx}\right)^2 + \sin\left(\dfrac{dy}{dx}\right) + 1 = 0\).

Hint:

Order = highest derivative present. Degree needs the equation to be a polynomial in the derivatives; sin(dy/dx) breaks that.

Answer 1

Step 1: Understanding the Concept:
Order is the highest derivative present. Degree is the power of the highest-order derivative, but degree is only defined if the equation can be written as a polynomial in ALL the derivatives that appear (no derivative sitting inside a trig, log, exponential, or root).

Step 2: Finding the order:
The highest derivative present is \(\dfrac{d^2y}{dx^2}\), so the order is 2.

Step 3: Checking whether degree is defined:
The term \(\sin\left(\dfrac{dy}{dx}\right)\) has a derivative, \(\dfrac{dy}{dx}\), sitting inside a sine function. A sine of a derivative cannot be expanded as a finite power of that derivative, so the whole left side is not a polynomial in the derivatives.

Final Answer:
The order is 2 and the degree is not defined. \[ \boxed{\text{Order} = 2,\ \text{Degree not defined}} \]