The solution of the differential equation ` x sin d (dy)/(dx) + ( x cos x + sin x ) y = sinx `. When `y (0)=0` is

Options

A. ` xy sin x = 1- cos x `
B. `xy sin x + cos x =0`
C. `x sin x + y cos x =0`
D. `x sin x + y cos x =1 `

Answer

To solve the differential equation

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

with the initial condition \( y(0) = 0 \), we will follow these steps:

### Step 1: Rearranging the equation
We start by rearranging the equation to isolate \(\frac{dy}{dx}\):

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

### Step 2: Simplifying the equation
This simplifies to:

\[ \frac{dy}{dx} = \frac{x \cos x + \sin x}{x \sin x} y + \frac{\sin x}{x \sin x} \]

This can be rewritten as:

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

### Step 3: Identifying \(p(x)\) and \(q(x)\)
Here, we identify:

- \(p(x) = \cot x + \frac{1}{x}\)
- \(q(x) = \frac{1}{x}\)

### Step 4: Finding the integrating factor
The integrating factor \(IF\) is given by:

\[ IF = e^{\int p(x) \, dx} \]

Calculating the integral:

\[ \int p(x) \, dx = \int \left(\cot x + \frac{1}{x}\right) \, dx = \ln(\sin x) + \ln(x) = \ln(x \sin x) \]

Thus, the integrating factor is:

\[ IF = e^{\ln(x \sin x)} = x \sin x \]

### Step 5: Multiplying through by the integrating factor
We multiply the entire differential equation by the integrating factor:

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

This can be expressed as:

\[ \frac{d}{dx}(x \sin x \cdot y) = \sin x \]

### Step 6: Integrating both sides
Integrating both sides with respect to \(x\):

\[ \int \frac{d}{dx}(x \sin x \cdot y) \, dx = \int \sin x \, dx \]

This gives:

\[ x \sin x \cdot y = -\cos x + C \]

### Step 7: Solving for \(y\)
Now, we solve for \(y\):

\[ y = \frac{-\cos x + C}{x \sin x} \]

### Step 8: Applying the initial condition
Using the initial condition \(y(0) = 0\):

As \(x \to 0\), \(-\cos(0) + C = C - 1\). The limit of \(\frac{-\cos x + C}{x \sin x}\) as \(x \to 0\) must equal \(0\). Thus, we find \(C = 1\).

### Final Solution
Substituting \(C = 1\) back into the equation gives:

\[ y = \frac{1 - \cos x}{x \sin x} \]