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

Options

A. `x = y^(2) + cy`
B. `x = 2 cy^(2)`
C. `x = 2y^(2) + cy`
D. None of these

Answer

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

### Step 1: Rearrange the equation
We start with the given equation:

\[ y \, dx - (x + 2y^2) \, dy = 0 \]

We can rearrange it to isolate \( dx \) and \( dy \):

\[ y \, dx = (x + 2y^2) \, dy \]

### Step 2: Separate variables
Next, we can separate the variables by dividing both sides by \( y(x + 2y^2) \):

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

### Step 3: Integrate both sides
Now, we integrate both sides:

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

The left side integrates to:

\[ \ln |x + 2y^2| + C_1 \]

And the right side integrates to:

\[ \ln |y| + C_2 \]

### Step 4: Combine the constants
We can combine the constants of integration:

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

### Step 5: Exponentiate both sides
To eliminate the logarithm, we exponentiate both sides:

\[ |x + 2y^2| = e^{\ln |y| + C} = |y| e^C \]

Let \( k = e^C \), a positive constant:

\[ |x + 2y^2| = k |y| \]

### Step 6: Solve for x
Now we can express \( x \) in terms of \( y \):

\[ x + 2y^2 = k y \quad \text{or} \quad x = k y - 2y^2 \]

### Step 7: General solution
Thus, the general solution of the differential equation is:

\[ x = 2y^2 + C y \]

where \( C \) is a constant.