Find the particular solution of the following :
`(dy)/(dx)= (x(2 log x + 1))/(sin y + y cos y)`, given that `y= pi/2`, when `x=1`.

Options

A. `y sin y = x^(2)log x + pi/2`
B. ` sin y = x^(2)log x + pi/2`
C. `y sin y = x^(3)log x + pi/2`
D. `y sin y = x^(2)log x + pi/6`

Answer

To find the particular solution of the differential equation

\[ \frac{dy}{dx} = \frac{x(2 \log x + 1)}{\sin y + y \cos y} \]

given that \( y = \frac{\pi}{2} \) when \( x = 1 \), we will follow these steps:

### Step 1: Separate the Variables
We start by rearranging the equation to separate variables \( y \) and \( x \):

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

### Step 2: Integrate Both Sides
Now we will integrate both sides:

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

### Step 3: Solve the Left Side Integral
For the left side, we can break it down:

1. The integral of \( \sin y \) is \( -\cos y \).
2. For \( y \cos y \), we use integration by parts:
- Let \( u = y \) and \( dv = \cos y \, dy \).
- Then \( du = dy \) and \( v = \sin y \).

Using integration by parts:

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

Thus, the left side becomes:

\[ -\cos y + (y \sin y + \cos y) = y \sin y \]

### Step 4: Solve the Right Side Integral
Now for the right side:

1. The integral of \( 2x \log x \) can also be solved using integration by parts:
- Let \( u = \log x \) and \( dv = 2x \, dx \).
- Then \( du = \frac{1}{x} \, dx \) and \( v = x^2 \).

Using integration by parts:

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

2. The integral of \( x \) is \( \frac{x^2}{2} \).

Thus, the right side becomes:

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

### Step 5: Combine Results
Now we can combine both sides:

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

### Step 6: Find the Constant \( C \)
We use the initial condition \( y = \frac{\pi}{2} \) when \( x = 1 \):

\[ \frac{\pi}{2} \sin\left(\frac{\pi}{2}\right) = 1^2 \log(1) + C \]

Since \( \sin\left(\frac{\pi}{2}\right) = 1 \) and \( \log(1) = 0 \):

\[ \frac{\pi}{2} = 0 + C \implies C = \frac{\pi}{2} \]

### Final Particular Solution
Substituting \( C \) back into the equation gives us:

\[ y \sin y = x^2 \log x + \frac{\pi}{2} \]

This is the particular solution to the given differential equation.

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