`(dy)/(dx) =-(y+sinx)/(x)` satisfying condition `y(0)=1`

Explanation

To solve the differential equation \(\frac{dy}{dx} = -\frac{y + \sin x}{x}\) with the initial condition \(y(0) = 1\), we can follow these steps:

### Step 1: Rearranging the Equation
Start by rearranging the given differential equation:
\[ \frac{dy}{dx} = -\frac{y + \sin x}{x} \]
Multiply both sides by \(x\) to eliminate the fraction:
\[ x \frac{dy}{dx} = -(y + \sin x) \]
This can be rewritten as:
\[ -x \frac{dy}{dx} = y + \sin x \]

### Step 2: Moving Terms
Now, we can rearrange the equation to isolate the terms involving \(y\):
\[ -x \frac{dy}{dx} - y = \sin x \]
This can be rearranged to:
\[ -x \frac{dy}{dx} = y + \sin x \]

### Step 3: Recognizing the Product Rule
Notice that the left-hand side can be expressed as a derivative:
\[ \frac{d}{dx}(xy) = x \frac{dy}{dx} + y \]
Thus, we can rewrite the equation as:
\[ \frac{d}{dx}(xy) = -\sin x \]

### Step 4: Integrating Both Sides
Now, integrate both sides:
\[ \int \frac{d}{dx}(xy) \, dx = \int -\sin x \, dx \]
This gives:
\[ xy = \cos x + C \]
where \(C\) is the constant of integration.

### Step 5: Applying the Initial Condition
We know from the problem that \(y(0) = 1\). Substitute \(x = 0\) and \(y = 1\) into the equation:
\[ 0 \cdot 1 = \cos(0) + C \]
This simplifies to:
\[ 0 = 1 + C \]
Thus, we find:
\[ C = -1 \]

### Step 6: Final Solution
Substituting \(C\) back into the equation gives:
\[ xy = \cos x - 1 \]
or rearranging it:
\[ y = \frac{\cos x - 1}{x} \]

### Final Answer
The solution to the differential equation is:
\[ y = \frac{\cos x - 1}{x} \]