Step 1: Understanding the Concept:
The given problem is a first-order differential equation. We observe that the variables \(x\) and \(y\) can be separated easily to find the general solution. Step 2: Key Formula or Approach:
We use the method of separation of variables:
\[ \int f(y) dy = \int g(x) dx \]
Integration of \(\frac{1}{y+1}\) is \(\ln(y+1)\) and the substitution method is used for the \(x\) integral. Step 3: Detailed Explanation:
Given equation: \(\left( \frac{2+\sin x}{y+1} \right) \frac{dy}{dx} = -\cos x\)
Separating the variables:
\[ \frac{dy}{y+1} = \frac{-\cos x}{2+\sin x} dx \]
Integrating both sides:
\[ \int \frac{1}{y+1} dy = -\int \frac{\cos x}{2+\sin x} dx \]
Let \(2+\sin x = t \implies \cos x dx = dt\).
\[ \ln(y+1) = -\ln(2+\sin x) + \ln C \]
\[ \ln(y+1) + \ln(2+\sin x) = \ln C \implies (y+1)(2+\sin x) = C \]
Using \(y(0) = 1\):
\[ (1+1)(2+\sin 0) = C \implies 2 \times 2 = C \implies C = 4 \]
So, the equation is \((y+1)(2+\sin x) = 4\).
Now, at \(x = \frac{\pi}{2}\):
\[ (y+1)(2 + \sin \frac{\pi}{2}) = 4 \implies (y+1)(2+1) = 4 \]
\[ 3(y+1) = 4 \implies y+1 = \frac{4}{3} \implies y = \frac{1}{3} \]
Step 4: Final Answer:
The value of \(y(\pi/2)\) is \(\frac{1}{3}\).