Question

Write the degree of the differential equation \( \frac{d}{dx} e^y + \frac{dy}{dx} = x \).

Hint:

To find the degree, ensure the differential equation is polynomial in derivatives; exponential terms like \( e^y \) do not affect the degree.

Answer 1

Rewrite the equation: \[ \frac{d}{dx} (e^y) + \frac{dy}{dx} = x. \] Since \( \frac{d}{dx} (e^y) = e^y \cdot \frac{dy}{dx} \), the equation becomes: \[ e^y \cdot \frac{dy}{dx} + \frac{dy}{dx} = x \quad \Rightarrow \quad \frac{dy}{dx} (e^y + 1) = x. \] The degree of a differential equation is the power of the highest-order derivative when the equation is polynomial in derivatives. Here, the highest-order derivative is \( \frac{dy}{dx} \), with power 1. Thus, the degree is: \[ \text{Degree} = 1. \]