Integrating factor of `x(dy)/(dx) - y = x^(4) - 3x` is

Options

A. x
B. log x
C. `(1)/(x)`
D. `-x`

Answer

To find the integrating factor of the differential equation \( x \frac{dy}{dx} - y = x^4 - 3x \), we will follow a systematic approach.

### Step 1: Rewrite the Equation
First, we need to rewrite the given equation in the standard form of a first-order linear differential equation, which is:
\[ \frac{dy}{dx} + P(x)y = Q(x) \]
To do this, we divide the entire equation by \( x \):
\[ \frac{dy}{dx} - \frac{y}{x} = x^3 - 3 \]

### Step 2: Identify \( P(x) \)
From the rewritten equation, we can identify \( P(x) \):
\[ P(x) = -\frac{1}{x} \]

### Step 3: Calculate the Integrating Factor
The integrating factor \( \mu(x) \) is given by the formula:
\[ \mu(x) = e^{\int P(x) \, dx} \]
Substituting \( P(x) \):
\[ \mu(x) = e^{\int -\frac{1}{x} \, dx} \]

### Step 4: Evaluate the Integral
Now we evaluate the integral:
\[ \int -\frac{1}{x} \, dx = -\ln|x| = \ln|x^{-1}| \]
Thus, we have:
\[ \mu(x) = e^{\ln|x^{-1}|} = |x^{-1}| \]
Since \( x \) is positive in the context of this problem, we can simplify this to:
\[ \mu(x) = \frac{1}{x} \]

### Step 5: Conclusion
Therefore, the integrating factor of the given differential equation is:
\[ \mu(x) = \frac{1}{x} \]