`(xcos(y/x))(dy)/(dx)=(ycos(y/x))+x`

Answer

To solve the given differential equation

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

we will follow these steps:

### Step 1: Rewrite the equation
We start by rewriting the equation in a more manageable form. We can divide both sides by \(x \cos(y/x)\) (assuming \(x \neq 0\) and \(\cos(y/x) \neq 0\)):

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

This simplifies to:

\[ \frac{dy}{dx} = \frac{y}{x} + \frac{1}{\cos(y/x)}. \]

### Step 2: Introduce a substitution
Let \(v = \frac{y}{x}\). Then, \(y = vx\) and \(\frac{dy}{dx} = v + x \frac{dv}{dx}\) (using the product rule). Substituting these into the equation gives:

\[ v + x \frac{dv}{dx} = v + \frac{1}{\cos(v)}. \]

### Step 3: Simplify the equation
Now, we can cancel \(v\) from both sides:

\[ x \frac{dv}{dx} = \frac{1}{\cos(v)}. \]

### Step 4: Separate variables
We can separate the variables to integrate:

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

### Step 5: Integrate both sides
Now, we integrate both sides:

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

The left side integrates to \(\sin(v)\) and the right side integrates to \(\log|x| + C\):

\[ \sin(v) = \log|x| + C. \]

### Step 6: Substitute back for \(v\)
Recall that \(v = \frac{y}{x}\), so we substitute back:

\[ \sin\left(\frac{y}{x}\right) = \log|x| + C. \]

### Final Solution
Thus, the final solution to the differential equation is:

\[ \sin\left(\frac{y}{x}\right) = \log|x| + C. \]

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