`x(dy)/(dx)=y-xcos^(2)(y/x)`

Answer

To solve the given differential equation \( x \frac{dy}{dx} = y - x \cos^2\left(\frac{y}{x}\right) \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ x \frac{dy}{dx} = y - x \cos^2\left(\frac{y}{x}\right) \]
We can divide both sides by \( x \) to isolate \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{y}{x} - \cos^2\left(\frac{y}{x}\right) \]

### Step 2: Identify the homogeneous function
The function on the right side can be expressed in terms of \( \frac{y}{x} \). Let \( v = \frac{y}{x} \). Then, we can rewrite the equation as:
\[ \frac{dy}{dx} = v - \cos^2(v) \]

### Step 3: Differentiate \( y \) in terms of \( v \)
Since \( y = vx \), we differentiate both sides with respect to \( x \):
\[ \frac{dy}{dx} = v + x \frac{dv}{dx} \]
Now, substituting this back into our equation gives:
\[ v + x \frac{dv}{dx} = v - \cos^2(v) \]

### Step 4: Simplify the equation
We can cancel \( v \) from both sides:
\[ x \frac{dv}{dx} = -\cos^2(v) \]

### Step 5: Separate variables
Now we can separate the variables:
\[ \frac{dv}{\cos^2(v)} = -\frac{dx}{x} \]

### Step 6: Integrate both sides
Integrating both sides gives:
\[ \int \sec^2(v) dv = -\int \frac{dx}{x} \]
The integral of \( \sec^2(v) \) is \( \tan(v) \), and the integral of \( -\frac{1}{x} \) is \( -\log|x| \):
\[ \tan(v) = -\log|x| + C \]

### Step 7: Substitute back for \( v \)
Recall that \( v = \frac{y}{x} \), so we substitute back:
\[ \tan\left(\frac{y}{x}\right) = -\log|x| + C \]

### Final Solution
Thus, the solution to the differential equation is:
\[ \tan\left(\frac{y}{x}\right) + \log|x| = C \]