Based on the following steps, rearrange the order to find the general solution of differential equation \( \cos x \frac{d y}{d x}+y \sin x=1 \). I \( \quad \) Integrating both sides \( \int \frac{d}{d x}(\sec x) y d x=\int \sec ^{2} x d x \) II \( \quad \) Multiply differential equation with integrating factor \( \sec x \frac{d y}{d x}+(\tan x) y \sec x=\sec ^{2} x \) III Identify \( P(x)=\tan x \) IV A Find the integrating factor, \( V(x)=e^{\int \tan x d x} \) B I, III, II and IV C II, III, IV and I D

To find the general solution of the differential equation \( \cos x \frac{d y}{d x}+y \sin x=1 \), we need to follow the correct order of steps. Let's analyze the steps provided: 1. **Identify the form of the equation**: The given equation can be rewritten as: \[ \frac{d y}{d x} + \tan x \cdot y = \sec^2 x \] This is a first-order linear differential equation of the form \( \frac{d y}{d x} + P(x) y = Q(x) \), where \( P(x) = \tan x \) and \( Q(x) = \sec^2 x \). 2. **Finding the integrating factor**: The integrating factor \( V(x) \) is given by: \[ V(x) = e^{\int P(x) \, dx} = e^{\int \tan x \, dx} \] 3. **Multiply the differential equation by the integrating factor**: This transforms the equation into a form that can be integrated: \[ V(x) \frac{d y}{d x} + V(x) P(x) y = V(x) Q(x) \] 4. **Integrate both sides**: After multiplying by the integrating factor, we can integrate both sides to find the solution. Now, let's rearrange the steps in the correct order: - **Step III**: Identify \( P(x) = \tan x \). - **Step IV**: Find the integrating factor, \( V(x) = e^{\int \tan x \, dx} \). - **Step II**: Multiply the differential equation with the integrating factor. - **Step I**: Integrate both sides. Thus, the correct order of steps is: **C: II, III, IV, and I**.