Question

Let g be a differentiable function such that $ \int_0^x g(t) dt = x - \int_0^x tg(t) dt $, $ x \ge 0 $ and let $ y = y(x) $ satisfy the differential equation $ \frac{dy}{dx} - y \tan x = 2(x+1) \sec x g(x) $, $ x \in \left[ 0, \frac{\pi}{2} \right) $. If $ y(0) = 0 $, then $ y\left( \frac{\pi}{3} \right) $ is equal to

• A. \( \frac{2\pi}{3\sqrt{3}} \)
• B. \( \frac{4\pi}{3} \)
• C. \( \frac{2\pi}{3} \)
• D. \( \frac{4\pi}{3\sqrt{3}} \)

Hint:

Recognize and solve the linear differential equation using the integrating factor method.

Answer 1

The Correct Option is B

Given:
\[\int_0^x g(t)\,dt = x - \int_0^x t g(t)\,dt\]Differentiating both sides with respect to \( x \) yields:\[g(x) = 1 - xg(x)\Rightarrow g(x)(1 + x) = 1\Rightarrow g(x) = \frac{1}{1 + x}\]Consider the differential equation:\[\frac{dy}{dx} - y \tan x = 2(x+1) \sec x \cdot g(x)= 2(x+1) \sec x \cdot \frac{1}{1+x} = 2 \sec x\]The equation simplifies to:\[\frac{dy}{dx} - y \tan x = 2 \sec x\]This is a linear differential equation. The integrating factor (IF) is calculated as:\[\text{IF} = e^{\int -\tan x \, dx} = e^{\ln|\cos x|} = \cos x\]Multiplying both sides by the integrating factor gives:\[\cos x \cdot \frac{dy}{dx} - y \cos x \tan x = 2\Rightarrow \frac{d}{dx}(y \cos x) = 2\]Integrating both sides yields:\[y \cos x = \int 2\,dx = 2x + C\]Applying the initial condition \( y(0) = 0 \):\[0 \cdot \cos 0 = 2 \cdot 0 + C \Rightarrow C = 0\]Consequently, the solution is:\[y \cos x = 2x \Rightarrow y = \frac{2x}{\cos x} = 2x \sec x\]Now, compute \( y\left( \frac{\pi}{3} \right) \):\[y\left( \frac{\pi}{3} \right) = 2 \cdot \frac{\pi}{3} \cdot \sec\left( \frac{\pi}{3} \right)= 2 \cdot \frac{\pi}{3} \cdot 2 = \frac{4\pi}{3}\]