Question

The general solution of the differential equation: $$ \cos^2 x \cdot \frac{dy}{dx} + y = \tan x $$ is $$ y = \tan x - 1 + C e^{-\tan x} $$ If this solution satisfies $ y\left(\frac{\pi}{4}\right) = 1 $, then find $ C $.

• A. \( e \)
• B. \( 1 \)
• C. \( -1 \)
• D. \( \frac{1}{e} \)

Hint:

When verifying a solution with an initial condition, plug the value into the general solution and solve for the constant.

Answer 1

The Correct Option is A

Given: \[ y = \tan x - 1 + C e^{-\tan x} \quad \text{and} \quad y\left(\frac{\pi}{4}\right) = 1 \] At \( x = \frac{\pi}{4} \Rightarrow \tan\left(\frac{\pi}{4}\right) = 1 \) Substitute into the solution: \[ 1 = 1 - 1 + C e^{-1} \Rightarrow 1 = C e^{-1} \Rightarrow C = e \]