Show that each of the following differential equations is homogeneous and solve each of them :
`(dy)/(dx)= y/x + x/y`.

Answer

To solve the differential equation \(\frac{dy}{dx} = \frac{y}{x} + \frac{x}{y}\), we will first show that it is a homogeneous differential equation and then solve it step by step.

### Step 1: Show that the equation is homogeneous

A differential equation is homogeneous if it can be expressed in the form \(f(tx, ty) = t^n f(x, y)\) for some \(n\).

Here, we rewrite the right-hand side of the equation:

\[ \frac{y}{x} + \frac{x}{y} \]

Substituting \(x\) with \(\lambda x\) and \(y\) with \(\lambda y\):

\[ f(\lambda x, \lambda y) = \frac{\lambda y}{\lambda x} + \frac{\lambda x}{\lambda y} = \frac{y}{x} + \frac{x}{y} \]

This shows that \(f(\lambda x, \lambda y) = f(x, y)\), which means the equation is homogeneous.

### Step 2: Substitute \(y = vx\)

Let \(y = vx\), where \(v\) is a function of \(x\). Then, we differentiate \(y\) with respect to \(x\):

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

### Step 3: Substitute into the differential equation

Now substitute \(y\) and \(\frac{dy}{dx}\) into the original equation:

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

This simplifies to:

\[ v + x\frac{dv}{dx} = v + \frac{1}{v} \]

### Step 4: Rearrange the equation

Subtract \(v\) from both sides:

\[ x\frac{dv}{dx} = \frac{1}{v} \]

### Step 5: Separate variables

Now, we can separate the variables:

\[ v \, dv = \frac{dx}{x} \]

### Step 6: Integrate both sides

Integrating both sides gives:

\[ \int v \, dv = \int \frac{dx}{x} \]

This results in:

\[ \frac{v^2}{2} = \ln |x| + C \]

### Step 7: Substitute back for \(y\)

Recall that \(v = \frac{y}{x}\), so we substitute back:

\[ \frac{1}{2} \left(\frac{y}{x}\right)^2 = \ln |x| + C \]

Multiplying through by \(2x^2\) gives:

\[ y^2 = 2x^2(\ln |x| + C) \]

### Final Solution

Thus, the solution to the differential equation is:

\[ y^2 = 2x^2 \ln |x| + 2Cx^2 \]