What is the general solution of the following differential equation `x^(2) dy + y^(2) dx = 0` ?

Options

A. `x + y = C `
B. `xy = C `
C. C( x+ y) = xy
D. None of these

Answer

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

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

### Step 2: Separating Variables
Next, we separate the variables \( y \) and \( x \):
\[ \frac{dy}{y^{2}} = -\frac{dx}{x^{2}} \]

### Step 3: Integrating Both Sides
Now we will integrate both sides:
\[ \int \frac{dy}{y^{2}} = \int -\frac{dx}{x^{2}} \]
The left-hand side integrates to:
\[ -\frac{1}{y} + C_1 \]
And the right-hand side integrates to:
\[ \frac{1}{x} + C_2 \]
Thus, we have:
\[ -\frac{1}{y} = \frac{1}{x} + C \]
where \( C = C_2 - C_1 \).

### Step 4: Rearranging the Equation
Rearranging the equation gives:
\[ \frac{1}{y} = -\frac{1}{x} - C \]
This can be rewritten as:
\[ \frac{1}{y} + \frac{1}{x} = -C \]

### Step 5: Final Form of the Solution
Multiplying through by \( xy \) to eliminate the fractions gives:
\[ x + y = -Cxy \]
Letting \( C_1 = -C \), we can write:
\[ x + y = C_1 xy \]
This is the general solution of the differential equation.

### Summary of the General Solution
The general solution of the differential equation \( x^{2} dy + y^{2} dx = 0 \) is:
\[ x + y = Cxy \]

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