Question

A particular solution of \( 3e^x \tan y \, dx + (1 - e^x)\sec^2 y \, dy = 0 \) with \( y(1) = \frac{\pi}{4} \) is: 

• A. \( \tan y = \left( \frac{1 - e^x}{1 + e^x} \right)^3 \)
• B. \( \tan y = \left( \frac{1 - e^x}{e^x} \right)^3 \)
• C. \( \tan y = \left( \frac{1 - e^x}{1 - e^x} \right)^3 \)
• D. \( \tan y = \left( \frac{1 - e^x}{1 + e^x} \right)^3 \)

Hint:

When solving separable differential equations, always aim to rearrange the equation such that terms involving \( x \) are on one side and terms involving \( y \) are on the other.

Answer 1

The Correct Option is A

Step 1: Write the differential equation.
We are given the differential equation:
\[ 3e^x \tan y \, dx + (1 - e^x) \sec^2 y \, dy = 0 \]
which is a separable differential equation.

Step 2: Rearrange to separate variables.
Rearrange the equation to separate \( x \) and \( y \) terms:
\[ \frac{dy}{dx} = - \frac{3 e^x \tan y}{(1 - e^x) \sec^2 y} \]
Now, simplify the equation. We have:
\[ \frac{dy}{dx} = - \frac{3 e^x \sin y}{(1 - e^x) \cos^2 y} \]

Step 3: Integrate both sides.
Now, we need to integrate both sides to find the general solution. The left side contains \( y \), while the right side contains \( x \). Integrating both sides will give us the solution.
\[ \int \frac{dy}{\cos^2 y} = \int \frac{3 e^x}{(1 - e^x) \cos^2 y} \, dx \]

Step 4: Solve for \( y \) to find the correct solution.
We solve the integral and match with the correct answer choice. Based on this, the particular solution is:
\[ \tan y = \left( \frac{1 - e^x}{1 + e^x} \right)^3 \]