The integrating factor of the differential equation :
`(dx)/(dy)+x/y=y` is :

Options

A. log y
B. y
C. `e^(y)`
D. None of these

Answer

To find the integrating factor of the differential equation

\[ \frac{dx}{dy} + \frac{x}{y} = y, \]

we can follow these steps:

### Step 1: Identify the standard form
The given equation can be rewritten in the standard form of a linear differential equation:

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

where \( P(y) = \frac{1}{y} \) and \( Q(y) = y \).

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

\[ \mu(y) = e^{\int P(y) \, dy}. \]

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

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

### Step 3: Calculate the integral
Now we compute the integral:

\[ \int \frac{1}{y} \, dy = \ln |y| + C, \]

where \( C \) is the constant of integration. Therefore,

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

Since \( y \) is positive in the context of this differential equation, we can simplify this to:

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

### Step 4: Conclusion
Thus, the integrating factor of the differential equation is:

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