Question

The solution of the differential equation \[ \frac{dy}{dx} = (4x + y + 1)^2 \] is:

• A. \( (4x + y + 1) = \tan(2x + C) \)
• B. \( (4x + y + 1) = 2 \tan(2x + C) \)
• C. \( (4x + y + 1) = 3 \tan(2x + C) \)
• D. \( (4x + y + 1) = 2 \tan(2x + C) \)

Hint:

To solve differential equations with separable variables, separate the terms and integrate both sides.

Answer 1

The Correct Option is D

Step 1: Separate variables and integrate.
We integrate the equation to find the general solution. The result will match the form \( (4x + y + 1) = 2 \tan(2x + C) \).
Step 2: Conclusion.
Thus, the correct solution is \( (4x + y + 1) = 2 \tan(2x + C) \).
Final Answer: \[ \boxed{2 \tan(2x + C)} \]