Step 1: Understanding the Concept:
The given differential equation is a first-order linear differential equation. An equation of the form can be solved using the integrating factor (I.F.) method.
Step 2: Key Formula or Approach:
1. Rewrite the equation in the standard form: .
2. Find the integrating factor: .
3. The solution is given by: .
Step 3: Detailed Explanation:
The given equation is .
First, we write it in the standard form by dividing by :
Here, and .
Next, we find the integrating factor:
Let , so . The integral becomes .
Now, the solution is:
We use the standard integration formula .
With , we get:
So, the solution is:
Isolating , we get:
Step 4: Final Answer:
The calculated solution matches the form of option (A), assuming the constant is an arbitrary constant of integration. The structure of the particular solution is identical.