The solution of the differential equation `(x+2y^(2))(dy)/(dx)=y`, is

Options

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

Answer

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

### Step 1: Rearranging the Equation
We start by rearranging the given equation:
\[ \frac{dy}{dx} = \frac{y}{x + 2y^2} \]

### Step 2: Inverting the Equation
Next, we can express \(\frac{dx}{dy}\) by inverting the equation:
\[ \frac{dx}{dy} = \frac{x + 2y^2}{y} \]

### Step 3: Separating Variables
Now, we can separate the variables. We rewrite the equation:
\[ dx - \frac{x}{y} dy = 2y dy \]
This can be rearranged as:
\[ \frac{dx}{dy} - \frac{x}{y} = 2y \]

### Step 4: Identifying the Linear Form
The equation is now in the standard linear form:
\[ \frac{dx}{dy} + P(y)x = Q(y) \]
where \(P(y) = -\frac{1}{y}\) and \(Q(y) = 2y\).

### Step 5: Finding the Integrating Factor
To solve this linear differential equation, we need to find the integrating factor, which is given by:
\[ \mu(y) = e^{\int P(y) \, dy} = e^{\int -\frac{1}{y} \, dy} = e^{-\ln|y|} = \frac{1}{y} \]

### Step 6: Multiplying the Equation by the Integrating Factor
We multiply the entire differential equation by the integrating factor:
\[ \frac{1}{y} \frac{dx}{dy} - \frac{x}{y^2} = 2 \]

### Step 7: 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 8: Integrating Both Sides
Now we integrate both sides with respect to \(y\):
\[ \int \frac{d}{dy}\left(\frac{x}{y}\right) dy = \int 2 \, dy \]
This gives:
\[ \frac{x}{y} = 2y + C \]
where \(C\) is the constant of integration.

### Step 9: Rearranging to Find \(x\)
Finally, we can express \(x\) in terms of \(y\):
\[ x = 2y^2 + Cy \]

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

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