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 can follow these steps:

### Step 1: Identify \( p(x) \) from the standard form
The equation can be rewritten in the standard form:

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

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

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

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

### Step 3: Evaluate the integral
Now, we compute the integral:

\[ \int \frac{1}{x} \, dx = \log |x|. \]

Thus, the integrating factor becomes:

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

Since \( x > 0 \), we can simplify this to:

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

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

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