Show that the following differential equations are homogeneous and solve them :
`x sec^(2)(y/x)dy={y sec^(2)(y/x)+x}dx`.

Answer

To show that the given differential equation is homogeneous and solve it, we start with the equation:

\[ x \sec^2\left(\frac{y}{x}\right) dy = \left(y \sec^2\left(\frac{y}{x}\right) + x\right) dx \]

### Step 1: Rewrite the equation

We can rearrange the equation to isolate \(dy\) and \(dx\):

\[ \frac{dy}{dx} = \frac{y \sec^2\left(\frac{y}{x}\right) + x}{x \sec^2\left(\frac{y}{x}\right)} \]

### Step 2: Check for homogeneity

To check if the function is homogeneous, we need to express it in the form \(f(kx, ky) = k^n f(x, y)\).

Let \(y = vx\) where \(v = \frac{y}{x}\). Then, \(dy = v dx + x dv\).

Substituting \(y = vx\) into the equation gives:

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

Now substituting into the right-hand side:

\[ \frac{dy}{dx} = \frac{vx \sec^2(v) + x}{x \sec^2(v)} = \frac{v \sec^2(v) + 1}{\sec^2(v)} \]

This simplifies to:

\[ \frac{dy}{dx} = v + \cos^2(v) \]

### Step 3: Substitute and simplify

Now we have:

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

Subtract \(v\) from both sides:

\[ x \frac{dv}{dx} = \cos^2(v) \]

### Step 4: Separate variables

Now we can separate the variables:

\[ \frac{dv}{\cos^2(v)} = \frac{dx}{x} \]

### Step 5: Integrate both sides

Integrate both sides:

\[ \int \sec^2(v) dv = \int \frac{dx}{x} \]

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

\[ \tan(v) = \ln|x| + C \]

### Step 6: Substitute back for \(v\)

Recall that \(v = \frac{y}{x}\):

\[ \tan\left(\frac{y}{x}\right) = \ln|x| + C \]

### Final Solution

Thus, the solution to the differential equation is:

\[ \tan\left(\frac{y}{x}\right) = \ln|x| + C \]