Question

The order and degree of the differential equation \((\frac{d^3y}{dx^3})^{\frac{2}{3}}-3\frac{d^2y}{dx^2}+5\frac{dy}{dx}+4 = 0\) are respectively

• A. \(2,3\)
• B. \(3,2\)
• C. \(3\), not defined
• D. not defined, \(3\)

Hint:

Clear the fractional power first, then read off the degree.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Order is the highest derivative present. Degree is the power of the highest-order derivative after the equation is made free of fractional powers of derivatives.

Step 2: Order:
The highest derivative is \(\frac{d^3y}{dx^3}\), so the order is 3.

Step 3: Degree:
Isolate the fractional term: \(\left(\frac{d^3y}{dx^3}\right)^{2/3} = 3\frac{d^2y}{dx^2} - 5\frac{dy}{dx} - 4\).
Cube both sides to remove the power \(\frac23\): \(\left(\frac{d^3y}{dx^3}\right)^2 = \left(3y'' - 5y' - 4\right)^3\).
The highest derivative now appears with power 2, so the degree is 2.

Step 4: Why the other options are wrong.
Degree 3 (A) comes from reading the 3 of the cube instead of the power of \(y'''\). "Not defined" does not apply because the equation can be made polynomial in derivatives.

Final Answer:
Order 3 and degree 2, option (B). \[ \boxed{3,\ 2} \]