Question

Let \( y(x) \) be the solution of the differential equation \[ \sqrt{\tan x}\, dy = \left(\sec^3 x - y (\tan x)^{3/2} \right)\, dx \] and \( y\left(\frac{\pi}{4}\right) = \frac{6\sqrt{2}}{5} \), then the value of \( y\left(\frac{\pi}{3}\right) \) is:

• A. \( \frac{8}{5} \cdot 3^{1/4} \)
• B. \( \frac{8}{3} \cdot 3^{1/4} \)
• C. \( \frac{8}{5} \cdot 5^{1/4} \)
• D. \( \frac{7}{5} \cdot 3^{1/4} \)

Hint:

Always simplify the differential equation first. Dividing by the coefficient of \( dy \) usually reveals a standard linear form or a Bernoulli form.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This is a first-order linear differential equation. We need to rearrange it into the standard form \( \frac{dy}{dx} + P(x)y = Q(x) \) and then find the Integrating Factor (I.F.).

Step 2: Key Formula or Approach:
1. Standard form: \( \frac{dy}{dx} + y \cdot \tan x = \frac{\sec^3 x}{\sqrt{\tan x}} \). 2. I.F. \( = e^{\int \tan x \, dx} = e^{\ln|\sec x|} = \sec x \).

Step 3: Detailed Explanation:
1. Rewrite the equation: \[ \frac{dy}{dx} + y \frac{(\tan x)^{3/2}}{\sqrt{\tan x}} = \frac{\sec^3 x}{\sqrt{\tan x}} \implies \frac{dy}{dx} + y \tan x = \frac{\sec^3 x}{\sqrt{\tan x}} \] 2. Multiply by I.F. (\( \sec x \)): \[ \frac{d}{dx}(y \sec x) = \frac{\sec^4 x}{\sqrt{\tan x}} \] 3. Integrate both sides: \[ y \sec x = \int \frac{(1 + \tan^2 x) \sec^2 x}{\sqrt{\tan x}} dx \] Let \( t = \tan x, dt = \sec^2 x \, dx \): \[ y \sec x = \int (t^{-1/2} + t^{3/2}) dt = 2\sqrt{t} + \frac{2}{5}t^{5/2} + C \] \[ y \sec x = 2\sqrt{\tan x} + \frac{2}{5}(\tan x)^{5/2} + C \] 4. Use \( y(\pi/4) = \frac{6\sqrt{2}}{5} \): \[ \frac{6\sqrt{2}}{5} \cdot \sqrt{2} = 2(1) + \frac{2}{5}(1) + C \implies \frac{12}{5} = \frac{12}{5} + C \implies C = 0 \] 5. Find \( y(\pi/3) \): \[ y \cdot 2 = 2(3^{1/4}) + \frac{2}{5}(3^{5/4}) = 2 \cdot 3^{1/4} + \frac{2 \cdot 3 \cdot 3^{1/4}}{5} = 3^{1/4} (2 + \frac{6}{5}) = \frac{16}{5} 3^{1/4} \] \[ y = \frac{8}{5} 3^{1/4} \]

Step 4: Final Answer:
The value of \( y(\pi/3) \) is \( \frac{8}{5} \cdot 3^{1/4} \).