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

To determine the order and degree of the given differential equation, we follow these steps:
1. Order of the differential equation: The order is determined by the highest derivative of $y$ that appears in the equation. In this case, the highest derivative is $\frac{d^2y}{dx^2}$, which is the second derivative of $y$ with respect to $x$. Therefore, the order of the equation is 2.
2. Degree of the differential equation: The degree of the equation is determined by the highest power of the highest derivative, after making sure that the equation is free from any irrational or fractional powers of the derivatives. However, the equation contains a logarithmic term, $ \log \left(\frac{d^2y}{dx^2}\right) $, involving a derivative. This makes the degree undefined because the logarithmic function cannot be expressed in polynomial form.
Hence, the order is 2, but the degree is not defined.