`x(dy)/(dx)-y+xsin(y/x)=0`

Explanation

To solve the homogeneous differential equation given by

\[ x \frac{dy}{dx} - y + x \sin\left(\frac{y}{x}\right) = 0, \]

we will follow a systematic approach.

### Step 1: Rearranging the Equation

We can rearrange the equation to isolate \(\frac{dy}{dx}\):

\[ x \frac{dy}{dx} = y - x \sin\left(\frac{y}{x}\right). \]

Dividing both sides by \(x\) (assuming \(x \neq 0\)) gives us:

\[ \frac{dy}{dx} = \frac{y}{x} - \sin\left(\frac{y}{x}\right). \]

### Step 2: Substitution

Let \(v = \frac{y}{x}\). Then, we have \(y = vx\). Differentiating \(y\) with respect to \(x\) using the product rule gives:

\[ \frac{dy}{dx} = v + x \frac{dv}{dx}. \]

### Step 3: Substituting into the Equation

Substituting \(y = vx\) and \(\frac{dy}{dx}\) into our rearranged equation:

\[ v + x \frac{dv}{dx} = v - \sin(v). \]

### Step 4: Simplifying the Equation

We can simplify this equation:

\[ x \frac{dv}{dx} = -\sin(v). \]

### Step 5: Separating Variables

We can separate the variables:

\[ \frac{dv}{-\sin(v)} = \frac{dx}{x}. \]

### Step 6: Integrating Both Sides

Now we integrate both sides:

\[ \int \frac{dv}{-\sin(v)} = \int \frac{dx}{x}. \]

The left side can be integrated as:

\[ -\ln|\csc(v) + \cot(v)| = \ln|x| + C, \]

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

### Step 7: Solving for \(y\)

From the integration, we can express \(v\) in terms of \(x\):

\[ \csc(v) + \cot(v) = \frac{1}{|x| e^{-C}}. \]

Substituting back \(v = \frac{y}{x}\), we can express \(y\) in terms of \(x\).

### Final Solution

The final solution will involve rearranging the above equation to find \(y\) explicitly in terms of \(x\).

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