Question

Identify the order and degree of the differential equation: \[ \left(\frac{d^3y}{dx^3}\right)^2 + 4\left(\frac{dy}{dx}\right)^4 + y = \sin(x) \]

• A. \( \text{Order } 3, \text{Degree } 4 \)
• B. \( \text{Order } 3, \text{Degree } 2 \)
• C. \( \text{Order } 4, \text{Degree } 3 \)
• D. \( \text{Order } 1, \text{Degree } 4 \)

Hint:

Degree depends only on the power of the highest order derivative, not on lower order derivatives.

Answer 1

The Correct Option is B

Concept:
• Order = Highest order derivative present in the differential equation.
• Degree = Power of the highest order derivative after removing radicals and fractions in derivatives.

Step 1: Identify the highest order derivative Given equation: \[ \left(\frac{d^3y}{dx^3}\right)^2 + 4\left(\frac{dy}{dx}\right)^4 + y = \sin(x) \] The derivatives present are: \[ \frac{dy}{dx} \quad \text{and} \quad \frac{d^3y}{dx^3} \] Among these, the highest order derivative is: \[ \frac{d^3y}{dx^3} \] Hence, \[ \boxed{\text{Order}=3} \]

Step 2: Determine the degree The highest order derivative appears as: \[ \left(\frac{d^3y}{dx^3}\right)^2 \] Therefore, its power is \(2\). Hence, \[ \boxed{\text{Degree}=2} \] Final Answer: \[ \boxed{\text{Order }3,\ \text{Degree }2} \]