Question

The order and degree of the differential equation \(3-(\frac{\text{d}^3y}{\text{d}x^3})^{\frac{7}{3}} = (\frac{\text{d}y}{\text{d}x})^5\) are respectively

• A. \(3,7\)
• B. \(7,3\)
• C. \(5,7\)
• D. \(3,5\)

Hint:

Clear the fractional power by isolating the radical term and raising both sides to a power.

Answer 1

The Correct Option is A

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 a polynomial in the derivatives (free of fractional powers).

Step 2: Clear the fraction
Isolate the term with the fractional power:
\[ \left(\frac{d^3y}{dx^3}\right)^{7/3} = 3 - \left(\frac{dy}{dx}\right)^5 \]
Cube both sides:
\[ \left(\frac{d^3y}{dx^3}\right)^{7} = \left[3 - \left(\frac{dy}{dx}\right)^5\right]^3 \]

Step 3: Read off
The highest derivative is the third derivative, so the order is 3. Its power is 7, so the degree is 7.
The pair is (3, 7), option (A). Option (D), (3, 5), takes the power of the first derivative, which is not the highest order.

Final Answer:
Order 3 and degree 7, option (A). \[ \boxed{3, 7} \]