Integrating factor of `(dy ) /( dx) +(y)/( x) = x^7 - 8 , ( x gt 0 ) ` is :

Options

A. x
B. log x
C. `-x`
D. `e^x`

Answer

To find the integrating factor for the differential equation

\[ \frac{dy}{dx} + \frac{y}{x} = x^7 - 8, \]

we will follow these steps:

### Step 1: Identify the form of the equation
The given equation is in the standard form of a first-order linear differential equation:

\[ \frac{dy}{dx} + P(x)y = Q(x), \]

where \( P(x) = \frac{1}{x} \) and \( Q(x) = x^7 - 8 \).

### Step 2: Find the integrating factor
The integrating factor \( \mu(x) \) is given by the formula:

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

Substituting \( P(x) = \frac{1}{x} \):

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

### Step 3: Calculate the integral
The integral \( \int \frac{1}{x} \, dx \) is:

\[ \int \frac{1}{x} \, dx = \ln |x| + C. \]

Thus, the integrating factor becomes:

\[ \mu(x) = e^{\ln |x|} = |x|. \]

Since \( x > 0 \), we can drop the absolute value:

\[ \mu(x) = x. \]

### Step 4: Conclusion
The integrating factor for the given differential equation is:

\[ \mu(x) = x. \]