Question

The degree of the differential equation \( x = \frac{dy}{dx} + \frac{1}{2!} \left(\frac{dy}{dx}\right)^2 + \frac{1}{3!} \left(\frac{dy}{dx}\right)^3 + \cdots \) is:

• A. 3
• B. 2
• C. 1
• D. Not defined

Hint:

The degree of a differential equation is the highest power of the highest derivative after rationalizing.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The infinite series is the expansion of \(e^{dy/dx} - 1\).

Step 2: Detailed Explanation:
\(x = \frac{dy}{dx} + \frac{1}{2!}\left(\frac{dy}{dx}\right)^2 + \frac{1}{3!}\left(\frac{dy}{dx}\right)^3 + \cdots = e^{dy/dx} - 1\).
So \(e^{dy/dx} = x + 1\).
\(\frac{dy}{dx} = \ln(x+1)\).
The differential equation is of first order and first degree.

Step 3: Final Answer:
Option (C) 1.