Show that each of the following differential equations is homogeneous and solve each of them :
`x^(2)(dy)/(dx)=y(x+y)`

Answer

To solve the differential equation \( x^2 \frac{dy}{dx} = y(x + y) \), we will first show that it is homogeneous and then solve it step by step.

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

A differential equation is homogeneous if it can be expressed in the form \( \frac{dy}{dx} = f\left(\frac{y}{x}\right) \).

Starting with the given equation:

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

We can rewrite it as:

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

Now, we can express \( \frac{y}{x} \) as \( v \), where \( y = vx \). Substituting \( y = vx \) into the equation gives:

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

Substituting \( y = vx \) into the right-hand side:

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

This confirms that the equation is homogeneous since both sides can be expressed in terms of \( v \) and \( x \).

### Step 2: Solve the differential equation

Now we will substitute \( y = vx \) into the equation and solve for \( v \).

1. Substitute \( y = vx \):

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

2. Simplifying gives:

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

3. Rearranging leads to:

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

4. Separating variables:

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

5. Integrating both sides:

\[ -\frac{1}{v} = -\frac{1}{x} + C \]

6. Rearranging gives:

\[ \frac{1}{v} = \frac{1}{x} + C \]

7. Substituting back \( v = \frac{y}{x} \):

\[ \frac{1}{\frac{y}{x}} = \frac{1}{x} + C \]

8. This leads to:

\[ \frac{x}{y} = \frac{1}{x} + C \]

9. Cross-multiplying gives:

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

10. Rearranging and simplifying leads to the general solution:

\[ y = \frac{x^2}{1 + Cx} \]

### Final Solution

Thus, the solution to the differential equation \( x^2 \frac{dy}{dx} = y(x + y) \) is:

\[ y = \frac{x^2}{1 + Cx} \]