If `x(dy)/(dx)=y(log y-log x+1)`, then the solution of the equation is

Options

A. `"log"(x)/(y)=cy`
B. `"log"(y)/(x)=cy`
C. `"log"(x)/(y)=cx`
D. `"log"(y)/(x)=cx`

Answer

To solve the differential equation \( x \frac{dy}{dx} = y (\log y - \log x + 1) \), we will follow these steps:

### Step 1: Rearranging the Equation
We start with the given equation:
\[ x \frac{dy}{dx} = y (\log y - \log x + 1) \]
We can rearrange it to isolate \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{y}{x} (\log y - \log x + 1) \]

### Step 2: Using Logarithmic Properties
Using the property of logarithms, we can rewrite \( \log y - \log x \) as \( \log \frac{y}{x} \):
\[ \frac{dy}{dx} = \frac{y}{x} \left( \log \frac{y}{x} + 1 \right) \]

### Step 3: Substituting \( y = vx \)
Let’s make the substitution \( y = vx \), where \( v \) is a function of \( x \). Then, we have:
\[ \frac{dy}{dx} = v + x \frac{dv}{dx} \]
Substituting this into our equation gives:
\[ v + x \frac{dv}{dx} = \frac{vx}{x} \left( \log v + 1 \right) \]
This simplifies to:
\[ v + x \frac{dv}{dx} = v (\log v + 1) \]

### Step 4: Rearranging the Equation
Now we can rearrange this equation:
\[ x \frac{dv}{dx} = v (\log v + 1 - 1) = v \log v \]
This can be expressed as:
\[ x \frac{dv}{dx} = v \log v \]

### Step 5: Separating Variables
We can separate the variables:
\[ \frac{dv}{v \log v} = \frac{dx}{x} \]

### Step 6: Integrating Both Sides
Now we integrate both sides:
\[ \int \frac{dv}{v \log v} = \int \frac{dx}{x} \]
The integral on the right side is straightforward:
\[ \int \frac{dx}{x} = \log x + C \]
For the left side, we use the substitution \( u = \log v \), which gives \( dv = v du \):
\[ \int \frac{1}{u} du = \log (\log v) + C_1 \]

### Step 7: Combining Results
After integration, we have:
\[ \log (\log v) = \log x + C \]
Exponentiating both sides gives:
\[ \log v = Cx \]
Exponentiating again results in:
\[ v = e^{Cx} \]

### Step 8: Substituting Back for \( y \)
Recall that \( y = vx \):
\[ y = e^{Cx} x \]
Thus, we can express the solution as:
\[ y = Cx e^{Cx} \]

### Final Solution
The solution of the differential equation is:
\[ y = Cx e^{Cx} \]