Solution of the differential `(x+2y^(3))=(dx)/(dy)y` is

Options

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

Explanation

To solve the differential equation \( x + 2y^3 = \frac{dx}{dy} y \), we will follow these steps:

### Step 1: Rearranging the Equation
Start by rewriting the given equation in a more manageable form:
\[ \frac{dx}{dy} = \frac{x + 2y^3}{y} \]

### Step 2: Simplifying the Right Side
We can separate the terms on the right-hand side:
\[ \frac{dx}{dy} = \frac{x}{y} + 2y^2 \]

### Step 3: Formulating the Linear Differential Equation
Now, we can rearrange this into the standard form of a linear differential equation:
\[ \frac{dx}{dy} - \frac{x}{y} = 2y^2 \]
Here, we identify \( p(y) = -\frac{1}{y} \) and \( q(y) = 2y^2 \).

### Step 4: Finding the Integrating Factor
The integrating factor \( \mu(y) \) is given by:
\[ \mu(y) = e^{\int p(y) \, dy} = e^{\int -\frac{1}{y} \, dy} \]
Calculating the integral:
\[ \int -\frac{1}{y} \, dy = -\ln|y| \implies \mu(y) = e^{-\ln|y|} = \frac{1}{y} \]

### Step 5: Multiplying Through by the Integrating Factor
Multiply the entire differential equation by the integrating factor:
\[ \frac{1}{y} \frac{dx}{dy} - \frac{x}{y^2} = 2 \]

### Step 6: Recognizing the Left Side as a Derivative
The left-hand side can be recognized as the derivative of a product:
\[ \frac{d}{dy}\left(\frac{x}{y}\right) = 2 \]

### Step 7: Integrating Both Sides
Integrate both sides with respect to \( y \):
\[ \int \frac{d}{dy}\left(\frac{x}{y}\right) dy = \int 2 \, dy \]
This gives us:
\[ \frac{x}{y} = 2y + C \]
where \( C \) is the constant of integration.

### Step 8: Solving for \( x \)
Now, multiply both sides by \( y \) to solve for \( x \):
\[ x = y(2y + C) = 2y^2 + Cy \]

### Final Solution
The solution of the differential equation is:
\[ x = 2y^2 + Cy \]