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Convert the following Homogenous differential equations to be separable

The given equation is:

\[
(x + y)dx - xdy = 0
\]

This is a homogeneous differential equation. To convert it to a separable form, we follow these steps:

### Step 1: Divide the entire equation by \(dx\)
\[
(x + y)dx - xdy = 0
\]
\[
(x + y) - x \frac{dy}{dx} = 0
\]

### Step 2: Isolate the derivative term
\[
x \frac{dy}{dx} = x + y
\]

### Step 3: Divide both sides by \(x\)
\[
\frac{dy}{dx} = 1 + \frac{y}{x}
\]

### Step 4: Use substitution \(v = \frac{y}{x}\), where \(y = vx\) and \(\frac{dy}{dx} = v + x\frac{dv}{dx}\)

Substitute \(y = vx\) into the equation:
\[
v + x\frac{dv}{dx} = 1 + v
\]

### Step 5: Simplify the equation
\[
x\frac{dv}{dx} = 1
\]

### Step 6: Separate the variables
\[
\frac{dv}{1} = \frac{dx}{x}
\]

Now the equation is separable.

### Step 7: Integrate both sides
\[
\int dv = \int \frac{dx}{x}
\]

\[
v = \ln |x| + C
\]

### Step 8: Substitute \(v = \frac{y}{x}\) back
\[
\frac{y}{x} = \ln |x| + C
\]

Thus, the solution to the original differential equation is:
\[
y = x(\ln |x| + C)
\]

This is the general solution.

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Would you like more details on any step?

### Related questions:
1. What are homogeneous differential equations?
2. How do you perform variable substitution in differential equations?
3. How does the method of separation of variables work in solving differential equations?
4. Can homogeneous differential equations always be solved using substitution?
5. What other methods can be used to solve first-order differential equations?

### Tip: Always check the type of differential equation (e.g., homogeneous, linear) before deciding on a solving method!

Convert the following homogeneous differential equation to be separable: (x + y)dx - xdy = 0

Learn how to convert the homogeneous differential equation (x + y)dx - xdy = 0 into a separable form using the substitution method and separation of variables. This step-by-step solution is ideal for university-level mathematics.

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