Write the particular solution of the differential equation : `(dy)/(dx)= sin x`, given that `y(pi)=2`.

Answer

To find the particular solution of the differential equation \(\frac{dy}{dx} = \sin x\) given that \(y(\pi) = 2\), we will follow these steps:

### Step 1: Rewrite the differential equation
We start with the given differential equation:
\[ \frac{dy}{dx} = \sin x \]

### Step 2: Separate variables
We can rewrite this equation as:
\[ dy = \sin x \, dx \]

### Step 3: Integrate both sides
Now, we integrate both sides:
\[ \int dy = \int \sin x \, dx \]
The left side integrates to \(y\), and the right side integrates to \(-\cos x + C\), where \(C\) is the constant of integration. Thus, we have:
\[ y = -\cos x + C \]

### Step 4: Use the initial condition to find \(C\)
We are given the condition \(y(\pi) = 2\). We substitute \(x = \pi\) and \(y = 2\) into our equation:
\[ 2 = -\cos(\pi) + C \]
We know that \(\cos(\pi) = -1\), so:
\[ 2 = -(-1) + C \]
This simplifies to:
\[ 2 = 1 + C \]
Subtracting 1 from both sides gives:
\[ C = 1 \]

### Step 5: Write the particular solution
Now that we have the value of \(C\), we can write the particular solution:
\[ y = -\cos x + 1 \]

### Final Answer:
The particular solution of the differential equation \(\frac{dy}{dx} = \sin x\) given that \(y(\pi) = 2\) is:
\[ y = -\cos x + 1 \]