To solve the differential equation \( x \cos y \frac{dy}{dx} + \sin y = x \) and find the curve that passes through the origin, we will follow these steps:
### Step 1: Rewrite the Differential Equation
We start with the given differential equation:
\[
x \cos y \frac{dy}{dx} + \sin y = x
\]
We can rearrange it to isolate \(\frac{dy}{dx}\):
\[
x \cos y \frac{dy}{dx} = x - \sin y
\]
\[
\frac{dy}{dx} = \frac{x - \sin y}{x \cos y}
\]
### Step 2: Separate Variables
Now, we can separate the variables \(y\) and \(x\):
\[
\frac{dy}{x - \sin y} = \frac{dx}{x \cos y}
\]
### Step 3: Integrate Both Sides
Next, we integrate both sides. The left side requires integrating with respect to \(y\) and the right side with respect to \(x\):
\[
\int \frac{dy}{x - \sin y} = \int \frac{dx}{x \cos y}
\]
This integration can be complex, but we can use the product rule for differentiation to simplify our work.
### Step 4: Use Product Rule
Using the product rule, we recognize that:
\[
\frac{d}{dx}(x \sin y) = x \frac{dy}{dx} + \sin y
\]
Thus, we can express our equation as:
\[
\frac{d}{dx}(x \sin y) = x
\]
### Step 5: Integrate the Product Rule Result
Now we can integrate:
\[
\int \frac{d}{dx}(x \sin y) = \int x \, dx
\]
This gives us:
\[
x \sin y = \frac{x^2}{2} + C
\]
### Step 6: Find the Constant \(C\)
Since the curve passes through the origin \((0, 0)\), we substitute \(x = 0\) and \(y = 0\):
\[
0 \cdot \sin(0) = \frac{0^2}{2} + C \implies 0 = 0 + C \implies C = 0
\]
### Step 7: Write the Particular Solution
Thus, the particular solution is:
\[
x \sin y = \frac{x^2}{2}
\]
or
\[
\sin y = \frac{x}{2}
\]
### Step 8: Check Points
Now we will check which points the curve passes through by substituting the \(x\) values from the options into the equation \(\sin y = \frac{x}{2}\).
1. **Option 1: (2, \(\frac{\pi}{2}\))**
\[
\sin y = \frac{2}{2} = 1 \implies y = \frac{\pi}{2} \quad \text{(Correct)}
\]
2. **Option 2: (-2, \(\frac{\pi}{2}\))**
\[
\sin y = \frac{-2}{2} = -1 \implies y = -\frac{\pi}{2} \quad \text{(Incorrect)}
\]
3. **Option 3: (4, y)**
\[
\sin y = \frac{4}{2} = 2 \quad \text{(Not possible, as sin cannot exceed 1)}
\]
4. **Option 4: (-8, y)**
\[
\sin y = \frac{-8}{2} = -4 \quad \text{(Not possible, as sin cannot be less than -1)}
\]
### Conclusion
The only point that the curve passes through, in addition to the origin, is:
\[
\boxed{(2, \frac{\pi}{2})}
\]
