Question

The order and degree of the differential equation

\[ \left[2\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 \right]^{\frac{3}{2}} = \frac{d^3y}{dx^3} \]

• A. 2 and 2
• B. 2 and 1
• C. 3 and 2
• D. 3 and 3
• E. 2 and 4

Hint:

Always remove radicals/fractions before determining degree.

Answer 1

The Correct Option is C

Understanding the Concept:
• Order = highest order derivative
• Degree = power of highest order derivative (after removing radicals)

Step 1: Identify order
Highest derivative: \[ \frac{d^3y}{dx^3} \] Thus order = 3

Step 2: Remove fractional power
\[ \left[2y'' + (y')^2\right]^{3/2} = y''' \] Square both sides: \[ \left[2y'' + (y')^2\right]^3 = (y''')^2 \]

Step 3: Identify degree
Highest order derivative = \(y'''\) Power = 2 Thus degree = 2 Final Answer: \[ \boxed{3 \text{ and } 2} \]