The solution of the differential equation `dy - (ydx)/(2x) = sqrt(x) ydy` is (where , c is an arbitrary constant)

Options

A. `y/(sqrtx) = y + c`
B. `y/(sqrtx) = (y^2)/2 + c`
C. `y = ysqrt(x) + c`
D. `y/(sqrtx) = -y^(2)+ c`

Answer

To solve the differential equation given by

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

we will follow these steps:

### Step 1: Rearranging the Equation
We start by rearranging the equation to isolate the terms involving \(dy\) and \(dx\):

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

### Step 2: Moving Terms
Next, we can move all terms involving \(dy\) to one side and \(dx\) to the other side:

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

### Step 3: Factoring Out \(dy\)
Now, we factor out \(dy\) from the left side:

\[ dy(1 - \sqrt{x}y) = \frac{y}{2x}dx. \]

### Step 4: Dividing by \(y\)
Assuming \(y \neq 0\), we can divide both sides by \(y\):

\[ \frac{dy}{y} = \frac{1}{2x(1 - \sqrt{x}y)}dx. \]

### Step 5: Integrating Both Sides
Now we integrate both sides. The left side integrates to \(\ln |y|\), and we will need to perform a substitution for the right side:

\[ \int \frac{dy}{y} = \int \frac{1}{2x(1 - \sqrt{x}y)}dx. \]

### Step 6: Simplifying the Right Side
Let's simplify the right side. We can rewrite it as:

\[ \frac{1}{2} \int \frac{1}{x(1 - \sqrt{x}y)}dx. \]

This integral can be solved using partial fractions or substitution methods.

### Step 7: Solving the Integral
After solving the integral on the right side, we will have:

\[ \ln |y| = \text{(result of the integral)} + C, \]

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

### Step 8: Exponentiating Both Sides
To solve for \(y\), we exponentiate both sides:

\[ y = e^{\text{(result of the integral)} + C} = e^{\text{(result of the integral)}} \cdot e^C. \]

Let \(k = e^C\), we can write:

\[ y = k \cdot e^{\text{(result of the integral)}}. \]

### Final Solution
Thus, the solution of the differential equation is:

\[ y = C \cdot e^{\text{(result of the integral)}}, \]

where \(C\) is an arbitrary constant.