Solve the following differential equations :
`(dy)/(dx)+y=x`.

Answer

To solve the differential equation \(\frac{dy}{dx} + y = x\), we will follow these steps:

### Step 1: Identify the standard form
The given equation is already in the standard form of a linear differential equation:
\[ \frac{dy}{dx} + P(x)y = Q(x) \]
where \(P(x) = 1\) and \(Q(x) = x\).

### Step 2: Find the integrating factor
The integrating factor \(I(x)\) is given by:
\[ I(x) = e^{\int P(x) \, dx} = e^{\int 1 \, dx} = e^{x} \]

### Step 3: Multiply the entire equation by the integrating factor
We multiply the entire differential equation by \(e^{x}\):
\[ e^{x} \frac{dy}{dx} + e^{x}y = e^{x}x \]

### Step 4: Recognize the left-hand side as a derivative
The left-hand side can be expressed as the derivative of a product:
\[ \frac{d}{dx}(e^{x}y) = e^{x}x \]

### Step 5: Integrate both sides
Now, we integrate both sides with respect to \(x\):
\[ \int \frac{d}{dx}(e^{x}y) \, dx = \int e^{x}x \, dx \]
The left side simplifies to:
\[ e^{x}y = \int e^{x}x \, dx \]
To solve the right side, we will use integration by parts. Let:
- \(u = x\) \(\Rightarrow du = dx\)
- \(dv = e^{x}dx\) \(\Rightarrow v = e^{x}\)

Using integration by parts:
\[ \int e^{x}x \, dx = uv - \int v \, du = xe^{x} - \int e^{x} \, dx = xe^{x} - e^{x} + C \]
where \(C\) is the constant of integration.

### Step 6: Substitute back
Now substituting back into our equation:
\[ e^{x}y = xe^{x} - e^{x} + C \]

### Step 7: Solve for \(y\)
To isolate \(y\), we divide both sides by \(e^{x}\):
\[ y = x - 1 + Ce^{-x} \]

### Final Solution
Thus, the solution to the differential equation is:
\[ y = x - 1 + Ce^{-x} \]