Solution of differential equation `(dy)/(dx)=xlogx` is

Options

A. `y=x^(2)log|x|-(x^(2))/(2)+c`
B. `y=(x^(2))/(2)log|x|-x^(2)+c`
C. `y=(x^(2))/(2)+(x^(2))/(2)log|x|+c`
D. `y=(x^(2))/(2)log|x|-(x^(2))/(4)+c`

Answer

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

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

### Step 2: Integrate both sides
Now, we will integrate both sides. The left side integrates to \(y\), and we need to integrate the right side:
\[ y = \int x \log x \, dx \]

### Step 3: Use integration by parts
To integrate \(\int x \log x \, dx\), we will use integration by parts. We choose:
- \(u = \log x\) (first function)
- \(dv = x \, dx\) (second function)

Now, we need to find \(du\) and \(v\):
- \(du = \frac{1}{x} \, dx\)
- \(v = \frac{x^2}{2}\)

Using the integration by parts formula \(\int u \, dv = uv - \int v \, du\), we have:
\[ \int x \log x \, dx = \left(\log x \cdot \frac{x^2}{2}\right) - \int \left(\frac{x^2}{2} \cdot \frac{1}{x}\right) dx \]
This simplifies to:
\[ = \frac{x^2}{2} \log x - \int \frac{x}{2} \, dx \]

### Step 4: Integrate the remaining integral
Now we need to integrate \(\int \frac{x}{2} \, dx\):
\[ \int \frac{x}{2} \, dx = \frac{1}{2} \cdot \frac{x^2}{2} = \frac{x^2}{4} \]

### Step 5: Combine the results
Now substituting back, we have:
\[ \int x \log x \, dx = \frac{x^2}{2} \log x - \frac{x^2}{4} + C \]
Thus, we can write:
\[ y = \frac{x^2}{2} \log x - \frac{x^2}{4} + C \]

### Step 6: Simplify the expression
To simplify, we can factor out \(\frac{x^2}{4}\):
\[ y = \frac{x^2}{4} \left(2 \log x - 1\right) + C \]

### Final Solution
The solution to the differential equation \(\frac{dy}{dx} = x \log x\) is:
\[ y = \frac{x^2}{2} \log x - \frac{x^2}{4} + C \]

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