The solution of the differential equaton
`(dy)/(dx)=(x log x^(2)+x)/(sin y+ycos y)`, is

Options

A. `ysiny=x^(2)logx+C`
B. `ysiny=x^(2)+C`
C. `ysiny=x^(2)+logx+C`
D. `y sin y=xlogx+C`

Answer

To solve the differential equation

\[ \frac{dy}{dx} = \frac{x \log x^2 + x}{\sin y + y \cos y}, \]

we will follow these steps:

### Step 1: Rearranging the Equation
We start by multiplying both sides of the equation by \((\sin y + y \cos y)\):

\[ (\sin y + y \cos y) \, dy = (x \log x^2 + x) \, dx. \]

### Step 2: Integrating Both Sides
Next, we integrate both sides:

\[ \int (\sin y + y \cos y) \, dy = \int (x \log x^2 + x) \, dx. \]

### Step 3: Integrating the Left Side
For the left side, we can separate the integrals:

\[ \int \sin y \, dy + \int y \cos y \, dy. \]

The integral of \(\sin y\) is:

\[ -\cos y. \]

For \(\int y \cos y \, dy\), we can use integration by parts, where \(u = y\) and \(dv = \cos y \, dy\). Thus, \(du = dy\) and \(v = \sin y\):

\[ \int y \cos y \, dy = y \sin y - \int \sin y \, dy = y \sin y + \cos y. \]

Combining these results, we get:

\[ \int (\sin y + y \cos y) \, dy = -\cos y + (y \sin y + \cos y) = y \sin y. \]

### Step 4: Integrating the Right Side
Now, we integrate the right side:

\[ \int (x \log x^2 + x) \, dx. \]

We can simplify \(x \log x^2\) to \(2x \log x\). Thus, we need to compute:

\[ \int (2x \log x + x) \, dx. \]

Using integration by parts for \(\int 2x \log x \, dx\), let \(u = \log x\) and \(dv = 2x \, dx\). Then, \(du = \frac{1}{x} \, dx\) and \(v = x^2\):

\[ \int 2x \log x \, dx = x^2 \log x - \int x^2 \cdot \frac{1}{x} \, dx = x^2 \log x - \int x \, dx = x^2 \log x - \frac{x^2}{2}. \]

Now, integrating \(x\):

\[ \int x \, dx = \frac{x^2}{2}. \]

Combining these results, we have:

\[ \int (2x \log x + x) \, dx = x^2 \log x - \frac{x^2}{2} + \frac{x^2}{2} = x^2 \log x. \]

### Step 5: Equating Both Integrals
Now we equate the results from both sides:

\[ y \sin y = x^2 \log x + C, \]

where \(C\) is the constant of integration.

### Final Solution
Thus, the solution to the differential equation is:

\[ y \sin y = x^2 \log x + C. \]

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