Solve the differential equation: `(x+3y^2)(dy)/(dx)=y`

Explanation

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

### Step 1: Rearranging the Equation
We start by rearranging the given equation:
\[ (x + 3y^2) \frac{dy}{dx} = y \]
Dividing both sides by \(y\) and rearranging gives:
\[ \frac{dy}{dx} = \frac{y}{x + 3y^2} \]

### Step 2: Cross-Multiplying
Next, we cross-multiply to separate the variables:
\[ (x + 3y^2) dy = y dx \]
This can be rewritten as:
\[ dx - \frac{x}{y} dy = 3y dy \]

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

### Step 4: Finding the Integrating Factor
To solve this linear differential equation, we need to find the integrating factor \(I\):
\[ I = e^{\int r \, dy} = e^{\int -\frac{1}{y} \, dy} = e^{-\log |y|} = \frac{1}{y} \]

### Step 5: Multiplying through by the Integrating Factor
Now, we multiply the entire equation by the integrating factor:
\[ \frac{1}{y} \frac{dx}{dy} - \frac{x}{y^2} = 3 \]
This simplifies to:
\[ \frac{d}{dy}\left(\frac{x}{y}\right) = 3 \]

### Step 6: Integrating Both Sides
Next, we integrate both sides with respect to \(y\):
\[ \int \frac{d}{dy}\left(\frac{x}{y}\right) dy = \int 3 \, dy \]
This gives us:
\[ \frac{x}{y} = 3y + C \]
where \(C\) is the constant of integration.

### Step 7: Solving for \(x\)
Finally, we solve for \(x\):
\[ x = 3y^2 + Cy \]

### Final Solution
Thus, the solution to the differential equation is:
\[ x = 3y^2 + Cy \]

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