Question

If \[ (2+\sin x)\frac{dy}{dx}+(y+1)\cos x=0 \] and \( y(0)=1 \), then \( y\left(\frac{\pi}{2}\right) \) is equal to:

• A. \( -\dfrac{2}{3} \)
• B. \( -\dfrac{1}{3} \)
• C. \( \dfrac{4}{3} \)
• D. \( \dfrac{1}{3} \)

Hint:

Whenever a differential equation contains a fraction of the form \[ \frac{f'(x)}{f(x)} \] its integral becomes: \[ \int \frac{f'(x)}{f(x)}dx=\ln|f(x)|+C \] Recognizing this pattern makes separable differential equations much easier to solve.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This problem is a repeat of Question 3. It utilizes the first-order differential equation separation technique.
It reinforces the concept of moving all functions of \( y \) to the left and all functions of \( x \) to the right.
Step 2: Key Formula or Approach:
The equation structure leads to the general solution:
\[ (y+1)(2+\sin x) = C \]
Step 3: Detailed Explanation:
Following the same algebraic steps as earlier:
\[ \frac{dy}{y+1} = \frac{-\cos x dx}{2+\sin x} \]
Integrating:
\[ \ln(y+1) = -\ln(2+\sin x) + \ln C \]
\[ (y+1)(2+\sin x) = C \]
With \( y(0)=1 \):
\[ (1+1)(2+0) = C \implies C = 4 \]
At \( x = \pi/2 \):
\[ (y+1)(2+1) = 4 \implies 3(y+1) = 4 \implies y = 1/3 \]
Step 4: Final Answer:
The answer is \( 1/3 \).
This is Option (4).