IF `y' = y/x(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 given by

\[ \frac{dy}{dx} = \frac{y}{x} \left( \log y - \log x + 1 \right), \]

we will follow these steps:

### Step 1: Rewrite the equation
We start by rewriting the equation in a more manageable form. We know that \( y' = \frac{dy}{dx} \), so we can express the equation as:

\[ \frac{dy}{dx} = \frac{y}{x} \left( \log y - \log x + 1 \right). \]

### Step 2: Substitute \( y = vx \)
Next, we will use the substitution \( y = vx \), where \( v \) is a function of \( x \). Differentiating \( y \) with respect to \( x \) gives:

\[ \frac{dy}{dx} = v + x \frac{dv}{dx}. \]

### Step 3: Substitute into the equation
Now we substitute \( y = vx \) and \( \frac{dy}{dx} = v + x \frac{dv}{dx} \) into the original equation:

\[ v + x \frac{dv}{dx} = \frac{vx}{x} \left( \log(vx) - \log x + 1 \right). \]

This simplifies to:

\[ v + x \frac{dv}{dx} = v \left( \log v + \log x - \log x + 1 \right) = v \left( \log v + 1 \right). \]

### Step 4: Simplify the equation
Now we can simplify the equation further:

\[ v + x \frac{dv}{dx} = v \log v + v. \]

Subtracting \( v \) from both sides gives:

\[ x \frac{dv}{dx} = v \log v. \]

### Step 5: Separate variables
We can separate the variables:

\[ \frac{dv}{v \log v} = \frac{dx}{x}. \]

### Step 6: Integrate both sides
Now we integrate both sides. The left side requires a substitution. Let \( t = \log v \), then \( dv = v dt = e^t dt \):

\[ \int \frac{dv}{v \log v} = \int \frac{1}{t} dt = \log |t| + C_1 = \log |\log v| + C_1. \]

The right side integrates to:

\[ \int \frac{dx}{x} = \log |x| + C_2. \]

### Step 7: Combine results
Equating the two integrals gives:

\[ \log |\log v| = \log |x| + C, \]

where \( C = C_2 - C_1 \).

### Step 8: Exponentiate both sides
Exponentiating both sides results in:

\[ |\log v| = k |x|, \]

where \( k = e^C \).

### Step 9: Substitute back for \( v \)
Recalling that \( v = \frac{y}{x} \), we have:

\[ |\log \left( \frac{y}{x} \right)| = k |x|. \]

### Step 10: Final expression
This leads us to the final expression:

\[ \log \left( \frac{y}{x} \right) = Cx, \]

where \( C \) is a constant.

### Final Answer
Thus, the solution to the differential equation is:

\[ \log \left( \frac{y}{x} \right) = Cx. \]