Integrating factor of `(x+2y^(3))(dy)/(dx)=y^(2)` is

Options

A. `e^(((1)/(y)))`
B. `e^(-((1)/(y)))`
C. y
D. `(-1)/(y)`

Explanation

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

### Step 1: Rewrite the equation in standard form
We start with the given equation:
\[ (x + 2y^3) \frac{dy}{dx} = y^2 \]
We can rearrange this to the standard form:
\[ \frac{dy}{dx} + P(y) y = Q(y) \]
where \(P(y)\) and \(Q(y)\) are functions of \(y\).

Dividing both sides by \(x + 2y^3\):
\[ \frac{dy}{dx} = \frac{y^2}{x + 2y^3} \]
This can be rewritten as:
\[ \frac{dy}{dx} + \frac{x}{x + 2y^3} y = \frac{y^2}{x + 2y^3} \]

### Step 2: Identify \(P(y)\)
From the rearranged equation, we can identify:
\[ P(y) = \frac{x}{x + 2y^3} \]
and
\[ Q(y) = \frac{y^2}{x + 2y^3} \]

### Step 3: Find the integrating factor
The integrating factor \(\mu(y)\) is given by:
\[ \mu(y) = e^{\int P(y) \, dy} \]
In our case, we need to find \(P(y)\) in terms of \(y\) only.

### Step 4: Simplify and integrate
To find the integrating factor, we need to express \(P\) in terms of \(y\). Since \(P(y)\) is dependent on \(x\), we need to express it properly.

However, we can simplify the original equation:
Rearranging gives:
\[ \frac{dy}{dx} = \frac{y^2}{x + 2y^3} \]
This can be written as:
\[ \frac{dy}{dx} + \frac{-x}{x + 2y^3} y = 0 \]
Now, we can find \(P(y)\):
\[ P(y) = -\frac{x}{x + 2y^3} \]

### Step 5: Calculate the integrating factor
Now, we need to find:
\[ \mu(y) = e^{\int P(y) \, dy} = e^{\int -\frac{1}{y^2} \, dy} \]
Calculating the integral:
\[ \int -\frac{1}{y^2} \, dy = \frac{1}{y} \]
Thus, the integrating factor is:
\[ \mu(y) = e^{\frac{1}{y}} \]

### Final Result
The integrating factor of the given differential equation is:
\[ e^{\frac{1}{y}} \]