What is the general solution of the differential equation ` x^(2) dy =y^(2) dx =0` ?
where c is the constant of intergration.

Options

A. x+y=c
B. xy=c
C. c(x+y)=c
D. none of the above

Explanation

To solve the differential equation \( x^2 dy + y^2 dx = 0 \), we can follow these steps:

### Step 1: Rearranging the Equation
We start with the given equation:
\[ x^2 dy + y^2 dx = 0 \]
We can rearrange this to isolate \( dy \) and \( dx \):
\[ x^2 dy = -y^2 dx \]

### Step 2: Separating Variables
Now, we can separate the variables by dividing both sides by \( y^2 x^2 \):
\[ \frac{dy}{y^2} = -\frac{dx}{x^2} \]

### Step 3: Integrating Both Sides
Next, we integrate both sides:
\[ \int \frac{dy}{y^2} = \int -\frac{dx}{x^2} \]
The integral of \( \frac{dy}{y^2} \) is \( -\frac{1}{y} \) and the integral of \( -\frac{dx}{x^2} \) is \( \frac{1}{x} \). Thus, we have:
\[ -\frac{1}{y} = \frac{1}{x} + C \]
where \( C \) is the constant of integration.

### Step 4: Rearranging the Equation
Now, we can rearrange this equation to express it in a more standard form:
\[ \frac{1}{y} + \frac{1}{x} = C \]

### Step 5: Finding the General Solution
To express the solution in a more familiar form, we can multiply through by \( xy \):
\[ x + y = Cxy \]
This gives us the general solution of the differential equation.

### Final Form
The final form of the general solution is:
\[ Cxy - x - y = 0 \]
where \( C \) is a constant.