The curve satisfying the differential equation `(dx)/(dy) = (x + 2yx^2)/(y-2x^3)` and passing through (1, 0) is given by

Options

A. `x^(2) + y^(2) = 1`
B. `x^(2) + y^(2) + y/x = 1`
C. `y^2 - y/x - x^2 = -1`
D. `x^2 - y^2 = 1`

Answer

To solve the differential equation given by

\[ \frac{dx}{dy} = \frac{x + 2yx^2}{y - 2x^3} \]

and find the curve that passes through the point (1, 0), we can follow these steps:

### Step 1: Cross-Multiply
Starting with the given equation, we can cross-multiply to eliminate the fraction:

\[ y \, dx - 2x^3 \, dx = (x + 2yx^2) \, dy \]

### Step 2: Rearrange the Equation
Next, we can rearrange the terms to group them together:

\[ 2yx^2 \, dy + 2x^3 \, dx + x \, dy - y \, dx = 0 \]

### Step 3: Divide by \(x^2\)
Now, we divide the entire equation by \(x^2\):

\[ 2y \, dy + 2x \, dx + \frac{x \, dy - y \, dx}{x^2} = 0 \]

### Step 4: Recognize the Derivative
The term \(\frac{x \, dy - y \, dx}{x^2}\) can be recognized as the derivative of \(\frac{y}{x}\):

\[ 2y \, dy + 2x \, dx + d\left(\frac{y}{x}\right) = 0 \]

### Step 5: Integrate the Equation
Now, we can integrate the entire equation:

\[ \int (2y \, dy) + \int (2x \, dx) + \int d\left(\frac{y}{x}\right) = C \]

This gives us:

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

### Step 6: Find the Constant \(C\)
To find the constant \(C\), we use the point (1, 0):

\[ 0^2 + 1^2 + \frac{0}{1} = C \implies C = 1 \]

### Step 7: Write the Final Equation
Substituting \(C\) back into our equation, we have:

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

This is the curve that satisfies the differential equation and passes through the point (1, 0).

### Final Answer:
The curve is given by:

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

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