Solve the following differential equations
`y{x cos (y/x)+y sin (y/x)}dx- x{ y sin (y/x)-x cos (y/x)}dy=0`.

Answer

To solve the differential equation

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

we will follow these steps:

### Step 1: Rewrite the Equation
We start by rewriting the equation in a more manageable form. We can express it as:

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

### Step 2: Identify Homogeneous Function
Next, we check if the equation is homogeneous. We can see that both \(y\) and \(x\) appear in a way that allows us to consider the substitution \(v = \frac{y}{x}\). This means \(y = vx\).

### Step 3: Substitute and Differentiate
Differentiating \(y = vx\) gives us:

\[ dy = v dx + x dv. \]

Substituting \(y\) and \(dy\) into the original equation, we get:

\[ vx \left( x \cos(v) + vx \sin(v) \right) dx - x \left( vx \sin(v) - x \cos(v) \right) (v dx + x dv) = 0. \]

### Step 4: Simplify the Equation
After substituting and simplifying, we can cancel out common terms and rearrange the equation to isolate \(dv\) and \(dx\):

\[ \frac{dv}{dx} = \frac{v \cos(v) + v^2 \sin(v) - v^2 \sin(v) + v \cos(v)}{v \sin(v) - \cos(v)}. \]

### Step 5: Separate Variables
We can now separate the variables:

\[ \frac{v \sin(v) - \cos(v)}{2v \cos(v)} dv = dx/x. \]

### Step 6: Integrate Both Sides
Integrating both sides gives us:

\[ \int \left( \frac{1}{2} \tan(v) dv - \int \frac{\cos(v)}{2v \cos(v)} dv \right) = \int \frac{dx}{x}. \]

### Step 7: Solve the Integrals
The integrals can be solved to yield:

\[ \frac{1}{2} \log |\sec(v)| - \frac{1}{2} \log |v| = \log |x| + C. \]

### Step 8: Exponentiate to Solve for v
Exponentiating both sides leads to:

\[ \frac{\sec(v)}{v} = Kx^2, \]

where \(K = e^{2C}\).

### Step 9: Substitute Back for y
Substituting back \(v = \frac{y}{x}\) gives:

\[ \frac{\sec\left(\frac{y}{x}\right)}{\frac{y}{x}} = Kx^2. \]

### Step 10: Final Solution
Rearranging gives us the final solution:

\[ xy \cos\left(\frac{y}{x}\right) = k, \]

where \(k\) is a constant.