Find the general solution of the following :
`(dy)/(dx)= log x`.

Answer

To find the general solution of the differential equation \(\frac{dy}{dx} = \log x\), we will follow these steps:

### Step 1: Separate the Variables
We can rewrite the equation as:
\[ dy = \log x \, dx \]

### Step 2: Integrate Both Sides
Next, we integrate both sides:
\[ \int dy = \int \log x \, dx \]
The left side integrates to:
\[ y = \int \log x \, dx \]

### Step 3: Integration by Parts
To solve \(\int \log x \, dx\), we will use integration by parts. We choose:
- \(u = \log x\) (which means \(du = \frac{1}{x} \, dx\))
- \(dv = dx\) (which means \(v = x\))

Using the integration by parts formula:
\[ \int u \, dv = uv - \int v \, du \]
we substitute our choices:
\[ \int \log x \, dx = x \log x - \int x \cdot \frac{1}{x} \, dx \]
This simplifies to:
\[ \int \log x \, dx = x \log x - \int 1 \, dx \]

### Step 4: Complete the Integration
Now, we integrate \(\int 1 \, dx\):
\[ \int 1 \, dx = x \]
Putting it all together, we have:
\[ \int \log x \, dx = x \log x - x + C \]

### Step 5: Write the General Solution
Now we can write the general solution:
\[ y = x \log x - x + C \]

### Final Answer
Thus, the general solution of the differential equation \(\frac{dy}{dx} = \log x\) is:
\[ y = x \log x - x + C \]