The Integrating Factor of the differential equation
`x(dy)/(dx)-y=2x^2` is

Options

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

Answer

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

### Step 1: Rewrite the Equation
We start by rewriting the given equation in the standard form of a first-order linear differential equation:

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

### Step 2: Divide by \( x \)
Next, we divide the entire equation by \( x \) to isolate \( \frac{dy}{dx} \):

\[ \frac{dy}{dx} - \frac{y}{x} = 2x \]

### Step 3: Identify \( p \) and \( q \)
Now, we can identify \( p \) and \( q \) from the standard form \( \frac{dy}{dx} + p y = q \):

- Here, \( p = -\frac{1}{x} \)
- And \( q = 2x \)

### Step 4: Calculate the Integrating Factor
The integrating factor \( \mu(x) \) is given by the formula:

\[ \mu(x) = e^{\int p \, dx} \]

Substituting \( p = -\frac{1}{x} \):

\[ \mu(x) = e^{\int -\frac{1}{x} \, dx} \]

### Step 5: Evaluate the Integral
The integral of \( -\frac{1}{x} \) is:

\[ \int -\frac{1}{x} \, dx = -\ln |x| = \ln |x|^{-1} \]

### Step 6: Apply the Exponential Function
Now, substituting back into the expression for the integrating factor:

\[ \mu(x) = e^{-\ln |x|} = \frac{1}{|x|} \]

Since we are typically interested in \( x > 0 \) for simplicity, we can write:

\[ \mu(x) = \frac{1}{x} \]

### Final Answer
Thus, the integrating factor of the differential equation is:

\[ \frac{1}{x} \]

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