`x(dy)/(dx)=y(logy-logx+1)`

Explanation

To solve the differential equation \( x \frac{dy}{dx} = y (\log y - \log x + 1) \), we will follow a systematic approach.

### Step-by-Step Solution:

1. **Rewrite the Equation**:
Start with the given equation:
\[ x \frac{dy}{dx} = y (\log y - \log x + 1) \]

2. **Separate Variables**:
We can rearrange the equation to isolate \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{y}{x} (\log y - \log x + 1) \]
This indicates that the equation is homogeneous since it can be expressed in terms of \(y/x\).

3. **Substitution**:
Let \(y = vx\), where \(v\) is a function of \(x\). Then, differentiate \(y\) with respect to \(x\):
\[ \frac{dy}{dx} = v + x \frac{dv}{dx} \]

4. **Substituting Back**:
Substitute \(y = vx\) into the equation:
\[ v + x \frac{dv}{dx} = \frac{vx}{x} (\log(vx) - \log x + 1) \]
Simplifying gives:
\[ v + x \frac{dv}{dx} = v (\log v + \log x - \log x + 1) = v (\log v + 1) \]

5. **Rearranging**:
Rearranging the equation leads to:
\[ x \frac{dv}{dx} = v (\log v + 1 - 1) = v \log v \]
Thus, we have:
\[ x \frac{dv}{dx} = v \log v \]

6. **Separating Variables Again**:
Separate the variables:
\[ \frac{dv}{v \log v} = \frac{dx}{x} \]

7. **Integrating Both Sides**:
Integrate both sides:
\[ \int \frac{dv}{v \log v} = \int \frac{dx}{x} \]
For the left side, use the substitution \(t = \log v\), which gives \(dv = e^t dt\):
\[ \int \frac{1}{t} dt = \log |t| + C = \log |\log v| + C \]
The right side integrates to:
\[ \log |x| + C \]

8. **Equating the Integrals**:
Thus, we have:
\[ \log |\log v| = \log |x| + C \]

9. **Exponentiating**:
Exponentiate both sides to eliminate the logarithm:
\[ |\log v| = e^{\log |x| + C} = |x| e^C \]
Let \(k = e^C\), then:
\[ \log v = kx \quad \text{or} \quad \log v = -kx \]

10. **Solving for \(v\)**:
This gives:
\[ v = e^{kx} \quad \text{or} \quad v = e^{-kx} \]

11. **Substituting Back for \(y\)**:
Recall that \(v = \frac{y}{x}\), thus:
\[ \frac{y}{x} = e^{kx} \implies y = x e^{kx} \]

### Final Solution:
The solution to the differential equation is:
\[ y = x e^{kx} \]
where \(k\) is a constant.