IF y = y (x) is the solution of the differential equation, `x (dy)/(dx) = y (log_(e) y - log_(e) x + 1)`, when y(1) = 2, then y(2) is equal to _______

Answer

To solve the differential equation \( x \frac{dy}{dx} = y (\log_e y - \log_e x + 1) \) with the initial condition \( y(1) = 2 \), we will follow these steps:

### Step 1: Rearranging the Equation
We start with the given differential equation:
\[ x \frac{dy}{dx} = y (\log_e y - \log_e x + 1) \]
Rearranging gives:
\[ \frac{dy}{dx} = \frac{y}{x} (\log_e y - \log_e x + 1) \]

### Step 2: Simplifying the Logarithmic Terms
Using the property of logarithms, we can combine the logarithmic terms:
\[ \log_e y - \log_e x + 1 = \log_e \left(\frac{y}{x}\right) + 1 \]
Thus, we can rewrite the equation as:
\[ \frac{dy}{dx} = \frac{y}{x} \left( \log_e \left(\frac{y}{x}\right) + 1 \right) \]

### Step 3: Substituting \( y = bx \)
To solve this homogeneous differential equation, we can substitute \( y = bx \), where \( b \) is a function of \( x \). Therefore:
\[ \frac{dy}{dx} = b + x \frac{db}{dx} \]
Substituting into the equation gives:
\[ b + x \frac{db}{dx} = b \left( \log_e b + 1 \right) \]

### Step 4: Rearranging the Equation
Rearranging leads to:
\[ x \frac{db}{dx} = b \log_e b \]
This can be separated as:
\[ \frac{db}{b \log_e b} = \frac{dx}{x} \]

### Step 5: Integrating Both Sides
Integrating both sides:
\[ \int \frac{db}{b \log_e b} = \int \frac{dx}{x} \]
Let \( t = \log_e b \), then \( db = b dt \) and \( b = e^t \), so:
\[ \int \frac{1}{t} dt = \int \frac{dx}{x} \]
This gives:
\[ \log_e |\log_e b| = \log_e |x| + C \]

### Step 6: Exponentiating
Exponentiating both sides results in:
\[ \log_e b = kx \]
where \( k = e^C \). Thus:
\[ b = e^{kx} \]

### Step 7: Substituting Back for \( y \)
Recalling that \( y = bx \):
\[ y = x e^{kx} \]

### Step 8: Using the Initial Condition
Using the initial condition \( y(1) = 2 \):
\[ 2 = 1 \cdot e^{k \cdot 1} \implies e^k = 2 \implies k = \log_e 2 \]
Thus, the solution becomes:
\[ y = x e^{(\log_e 2)x} = x \cdot 2^x \]

### Step 9: Finding \( y(2) \)
Now we need to find \( y(2) \):
\[ y(2) = 2 \cdot 2^2 = 2 \cdot 4 = 8 \]

### Final Answer
Thus, \( y(2) \) is equal to:
\[ \boxed{8} \]

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