Question

If $y = y(x)$ is the solution of $\left(\frac{2 + \sin x}{y + 1}\right) \frac{dy}{dx} + \cos x = 0$ with $y(0) = 1$, then $y\left(\frac{\pi}{2}\right) = $

• A. $\frac{1}{3}$
• B. $\frac{2}{3}$
• C. $1$
• D. $\frac{4}{3}$

Hint:

For separable equations, integrate both sides and apply the initial condition immediately to find the constant $C$.

Answer 1

The Correct Option is A

Step 1: Concept
This is a variable separable differential equation. We rearrange terms to group $y$ variables on one side and $x$ variables on the other.

Step 2: Meaning
Rearranging gives: $\frac{1}{y + 1} dy = \frac{-\cos x}{2 + \sin x} dx$. Integrating both sides: $\ln(y + 1) = -\ln(2 + \sin x) + \ln C$.

Step 3: Analysis
This simplifies to $(y + 1)(2 + \sin x) = C$. Using the initial condition $y(0) = 1$: $(1 + 1)(2 + \sin 0) = C \Rightarrow 2(2) = 4$, so $C = 4$. The general solution is $(y + 1)(2 + \sin x) = 4$.

Step 4: Conclusion
To find $y(\pi/2)$, substitute $x = \pi/2$: $(y + 1)(2 + \sin(\pi/2)) = 4 \Rightarrow (y + 1)(2 + 1) = 4 \Rightarrow 3(y + 1) = 4 \Rightarrow y + 1 = 4/3 \Rightarrow y = 1/3$. Final Answer: (A)