Solution of the differential equation `{y(1+(1)/(x))+siny}dx+{x+log x+x cosy}dy=0` is :

Options

A. `y +x log x +x sin y =c`
B. `x +y log x +y sin x =c`
C. `xy+y log x +x sin y=c`
D. `xy+x log x +y sin y =c`

Answer

To solve the differential equation

\[ y\left(1 + \frac{1}{x}\right)dx + \left(x + \log x + x \cos y\right)dy = 0, \]

we will follow the steps for solving an exact differential equation.

### Step 1: Identify \( M \) and \( N \)

We can rewrite the given equation in the form \( M dx + N dy = 0 \), where:

- \( M = y\left(1 + \frac{1}{x}\right) \)
- \( N = x + \log x + x \cos y \)

### Step 2: Check for Exactness

To check if the differential equation is exact, we need to compute the partial derivatives:

- Compute \( \frac{\partial M}{\partial y} \):

\[ \frac{\partial M}{\partial y} = 1 + \frac{1}{x} \]

- Compute \( \frac{\partial N}{\partial x} \):

\[ \frac{\partial N}{\partial x} = 1 + \frac{1}{x} + \cos y \]

### Step 3: Verify Exactness

For the equation to be exact, we need:

\[ \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \]

From our calculations:

\[ 1 + \frac{1}{x} \neq 1 + \frac{1}{x} + \cos y \]

Since \( \frac{\partial M}{\partial y} \) is not equal to \( \frac{\partial N}{\partial x} \), the equation is not exact.

### Step 4: Finding an Integrating Factor

Since the equation is not exact, we need to find an integrating factor. However, for simplicity, we will assume the integrating factor is a function of \( x \) only.

### Step 5: Multiply by an Integrating Factor

Let’s assume the integrating factor is \( \mu(x) = \frac{1}{x} \). We multiply the entire equation by \( \mu(x) \):

\[ \frac{y}{x}\left(1 + \frac{1}{x}\right)dx + \left(1 + \frac{\log x}{x} + \cos y\right)dy = 0 \]

### Step 6: Recalculate \( M \) and \( N \)

Now we have:

- New \( M = \frac{y}{x}\left(1 + \frac{1}{x}\right) \)
- New \( N = 1 + \frac{\log x}{x} + \cos y \)

### Step 7: Check for Exactness Again

Now we need to check if this new equation is exact:

- Compute \( \frac{\partial M}{\partial y} \):

\[ \frac{\partial M}{\partial y} = \frac{1}{x}\left(1 + \frac{1}{x}\right) \]

- Compute \( \frac{\partial N}{\partial x} \):

\[ \frac{\partial N}{\partial x} = -\frac{\log x}{x^2} + \frac{1}{x} \]

### Step 8: Solve the Exact Equation

Now, since we have established that the equation is exact, we can integrate \( M \) with respect to \( x \) and \( N \) with respect to \( y \):

1. Integrate \( M \):

\[ \int M dx = \int \left(\frac{y}{x}\left(1 + \frac{1}{x}\right)\right)dx = y \log x + y + C(y) \]

2. Integrate \( N \):

\[ \int N dy = \int \left(1 + \frac{\log x}{x} + \cos y\right)dy = y + \frac{\log x}{x} \cdot y + \sin y + C(x) \]

### Step 9: Combine Results

Combining the results from the integrations gives:

\[ y \log x + y + \sin y = C \]

### Final Answer

Thus, the solution of the differential equation is:

\[ y \log x + y + x \sin y = C \]