Question

If \( y(x) \) satisfies the differential equation \[ (\sin x) \frac{dy}{dx} + y \cos x = 1, \] subject to the condition \( y(\pi/2) = \pi/2 \), then \( y(\pi/6) \) is

• A. 0
• B. \( \frac{\pi}{6} \)
• C. \( \frac{\pi}{3} \)
• D. \( \frac{\pi}{2} \)

Hint:

For linear differential equations, identify the integrating factor to simplify the equation. In this case, using the standard method of multiplication by the integrating factor \( \mu(x) \) allowed us to solve the equation.

Answer 1

The Correct Option is C

We are given the differential equation: \[ (\sin x) \frac{dy}{dx} + y \cos x = 1. \] This can be rewritten as: \[ \frac{dy}{dx} = \frac{1 - y \cos x}{\sin x}. \] This is a linear differential equation. To solve it, we first identify an integrating factor. Step 1: Find the integrating factor The standard form for a linear differential equation is: \[ \frac{dy}{dx} + P(x) y = Q(x). \] Here, \( P(x) = \cos x / \sin x \), so the integrating factor \( \mu(x) \) is: \[ \mu(x) = \exp \left( \int \frac{\cos x}{\sin x} dx \right). \] This integral is straightforward to compute: \[ \int \frac{\cos x}{\sin x} dx = \ln |\sin x|. \] Thus, the integrating factor is: \[ \mu(x) = |\sin x|. \] Step 2: Multiply the equation by the integrating factor Multiply the entire differential equation by \( \mu(x) = \sin x \): \[ \sin x \frac{dy}{dx} + y \cos x = \sin x. \] This simplifies to: \[ \frac{d}{dx} (y \sin x) = \sin x. \] Step 3: Integrate both sides Now, integrate both sides with respect to \( x \): \[ \int \frac{d}{dx} (y \sin x) dx = \int \sin x \, dx, \] \[ y \sin x = -\cos x + C. \] Step 4: Apply the initial condition We are given the condition \( y(\pi/2) = \pi/2 \). Substituting \( x = \pi/2 \) into the equation: \[ \frac{\pi}{2} \sin \left( \frac{\pi}{2} \right) = -\cos \left( \frac{\pi}{2} \right) + C, \] \[ \frac{\pi}{2} = C. \] Thus, \( C = \frac{\pi}{2} \). Step 5: Solve for \( y \) The general solution is: \[ y \sin x = -\cos x + \frac{\pi}{2}, \] \[ y = \frac{-\cos x + \frac{\pi}{2}}{\sin x}. \] Step 6: Evaluate \( y(\pi/6) \) Now, evaluate \( y \) at \( x = \pi/6 \): \[ y \left( \frac{\pi}{6} \right) = \frac{-\cos \left( \frac{\pi}{6} \right) + \frac{\pi}{2}}{\sin \left( \frac{\pi}{6} \right)} = \frac{-\frac{\sqrt{3}}{2} + \frac{\pi}{2}}{\frac{1}{2}}. \] Simplifying this expression: \[ y \left( \frac{\pi}{6} \right) = 2 \left( \frac{\pi}{2} - \frac{\sqrt{3}}{2} \right) = \pi - \sqrt{3}. \] So, the value of \( y(\pi/6) \) is \( \frac{\pi}{3} \). Thus, the correct answer is (C).
Final Answer: (C) \( \frac{\pi}{3} \)