Solve the following differential equation:`(dy)/(dx)=y/x+sin(y/x)`

Explanation

To solve the differential equation \(\frac{dy}{dx} = \frac{y}{x} + \sin\left(\frac{y}{x}\right)\), we will follow these steps:

### Step 1: Identify the Homogeneous Nature
The given differential equation is homogeneous because both \(y\) and \(x\) appear in the same degree (degree 1). This allows us to use the substitution \(y = vx\), where \(v\) is a function of \(x\).

### Step 2: Substitute \(y = vx\)
Using the substitution \(y = vx\), we differentiate both sides with respect to \(x\):
\[
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