Question

If $y(x)$ satisfies the differential equation $(\sin x)\frac{dy}{dx} + y \cos x - 1 = 0$ subject to the condition $y\left(\frac{\pi}{2}\right) = 0$, then $y\left(\frac{\pi}{6}\right) = $_______

• A. $\frac{\pi}{3}$
• B. $\frac{-2\pi}{3}$
• C. $\frac{-\pi}{6}$
• D. $\frac{-\pi}{3}$

Hint:

Always look for exact differential forms before applying standard integrating factor methods for linear differential equations.
Recognizing the product rule pattern can save significant time during exams.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This question asks us to find the particular solution to a first-order ordinary differential equation with a given initial condition, and then evaluate $y(x)$ at $x = \frac{\pi}{6}$.

Step 2: Key Formula or Approach:
The given differential equation can be recognized as an exact derivative using the product rule of differentiation:
\[ \frac{d}{dx} [u(x) v(x)] = u(x) \frac{dv}{dx} + v(x) \frac{du}{dx} \]

Step 3: Detailed Explanation:

• Write down the given differential equation:
\[ (\sin x)\frac{dy}{dx} + y \cos x - 1 = 0 \]

• Rearranging the equation:
\[ (\sin x)\frac{dy}{dx} + y \cos x = 1 \]

• Observe that the left-hand side is the exact derivative of the product $y \sin x$ with respect to $x$:
\[ \frac{d}{dx} [y \sin x] = y \cos x + (\sin x) \frac{dy}{dx} \]

• Thus, the equation simplifies to:
\[ \frac{d}{dx} [y \sin x] = 1 \]

• Integrating both sides with respect to $x$:
\[ y \sin x = \int 1 dx \]
\[ y \sin x = x + C \]
where $C$ is the constant of integration.

• Now, we apply the given initial condition $y\left(\frac{\pi}{2}\right) = 0$ to find $C$:
\[ (0) \sin\left(\frac{\pi}{2}\right) = \frac{\pi}{2} + C \]
\[ 0 = \frac{\pi}{2} + C \implies C = -\frac{\pi}{2} \]

• Substituting the value of $C$ back into the solution:
\[ y \sin x = x - \frac{\pi}{2} \]

• Rearranging to solve for $y$:
\[ y(x) = \frac{x - \frac{\pi}{2}}{\sin x} \]

• Next, evaluate the function at $x = \frac{\pi}{6}$:
\[ y\left(\frac{\pi}{6}\right) = \frac{\frac{\pi}{6} - \frac{\pi}{2}}{\sin\left(\frac{\pi}{6}\right)} \]

• Simplify the numerator:
\[ \frac{\pi}{6} - \frac{\pi}{2} = \frac{\pi - 3\pi}{6} = \frac{-2\pi}{6} = -\frac{\pi}{3} \]

• The denominator value is $\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$.

• Calculate the final value:
\[ y\left(\frac{\pi}{6}\right) = \frac{-\frac{\pi}{3}}{\frac{1}{2}} = -\frac{2\pi}{3} \]

Step 4: Final Answer
The value of $y\left(\frac{\pi}{6}\right)$ is $-\frac{2\pi}{3}$, which corresponds to option (B).