Step 1: Understanding the Concept:
This is a first-order differential equation that can be solved using the method of separation of variables. This method involves rearranging the equation so that all terms involving 'y' are on one side with 'dy', and all terms involving 'x' are on the other side with 'dx'.
Step 2: Key Formula or Approach:
1. Separate the variables: Move all y-terms to the left and all x-terms to the right.
2. Integrate both sides of the equation.
3. Solve for y to find the general solution.
Step 3: Detailed Explanation:
The given differential equation is:
\[ \frac{dy}{dx} = 1 + y^2 \]
Separate the variables by multiplying by $dx$ and dividing by $1+y^2$:
\[ \frac{dy}{1+y^2} = dx \]
Now, integrate both sides:
\[ \int \frac{1}{1+y^2} \,dy = \int 1 \,dx \]
The integral of the left side is a standard form: $\int \frac{1}{1+u^2} \,du = \tan^{-1}(u)$.
The integral of the right side is straightforward.
\[ \tan^{-1}(y) = x + c \]
where 'c' is the constant of integration.
To solve for y, take the tangent of both sides:
\[ y = \tan(x + c) \]
Step 4: Final Answer:
The general solution of the differential equation is $y = \tan(x + c)$. Therefore, option (B) is correct.