Question

The order and degree of the differential equation \[ \frac{d^2y}{dx^2} + 4 \left(\frac{dy}{dx}\right) = x \log \left(\frac{d^2y}{dx^2}\right) \text{ are respectively:} \]

• A. 0, 3
• B. 2, 1
• C. 2, not defined
• D. 1, not defined

Hint:

When the equation contains a derivative inside a non-algebraic function (such as a logarithm), the degree is considered not defined.

Answer 1

The Correct Option is C

The order and degree of the differential equation are determined as follows:
1. Order: The order is the highest order of the derivative of $y$ present. The highest derivative is $\frac{d^2y}{dx^2}$, the second derivative. Thus, the order is 2.
2. Degree: The degree is the highest power of the highest derivative, after ensuring derivatives are free from irrational or fractional powers. The presence of $ \log \left(\frac{d^2y}{dx^2}\right) $ means the equation cannot be expressed polynomially in terms of its derivatives. Consequently, the degree is undefined.
Therefore, the order is 2, and the degree is undefined.