The general solution of `x((dy)/(dx))+(logx)y=x^(-1/2logx)` is

Options

A. `y = x^(1-1/2log x)+cx^(-1/2 log x)`
B. `y . x^(1/2 log x) = x^(1/2 log x) + c`
C. `y = e^(((log x)^(2))/(2))(x+c)`
D. `y = e^(1/2(log x)^(2)) (x^(1-(1)/(2)(log x))-x^(-(1)/(2)log x))+ c`

Explanation

To solve the differential equation \( x \frac{dy}{dx} + \log x \cdot y = x^{-\frac{1}{2} \log x} \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ x \frac{dy}{dx} + \log x \cdot y = x^{-\frac{1}{2} \log x} \]
We can divide through by \( x \) to simplify it:
\[ \frac{dy}{dx} + \frac{\log x}{x} \cdot y = x^{-\frac{1}{2} \log x - 1} \]

### Step 2: Identify \( p(x) \) and \( q(x) \)
From the standard form of a linear differential equation \( \frac{dy}{dx} + p(x) y = q(x) \), we identify:
\[ p(x) = \frac{\log x}{x}, \quad q(x) = x^{-\frac{1}{2} \log x - 1} \]

### Step 3: Find the integrating factor
The integrating factor \( \mu(x) \) is given by:
\[ \mu(x) = e^{\int p(x) \, dx} = e^{\int \frac{\log x}{x} \, dx} \]
Let \( t = \log x \), then \( dt = \frac{1}{x} dx \) or \( dx = x \, dt \). Thus, we have:
\[ \int \frac{\log x}{x} \, dx = \int t \, dt = \frac{t^2}{2} + C = \frac{(\log x)^2}{2} + C \]
So,
\[ \mu(x) = e^{\frac{(\log x)^2}{2}} = x^{\frac{1}{2} \log x} \]

### Step 4: Multiply through by the integrating factor
Multiplying the entire differential equation by the integrating factor:
\[ x^{\frac{1}{2} \log x} \frac{dy}{dx} + x^{\frac{1}{2} \log x} \cdot \frac{\log x}{x} \cdot y = x^{\frac{1}{2} \log x} \cdot x^{-\frac{1}{2} \log x - 1} \]
This simplifies to:
\[ x^{\frac{1}{2} \log x} \frac{dy}{dx} + \log x \cdot y \cdot x^{-\frac{1}{2} \log x} = x^{-1} \]

### Step 5: Recognize the left side as a derivative
The left-hand side can be recognized as the derivative of a product:
\[ \frac{d}{dx} \left( y \cdot x^{\frac{1}{2} \log x} \right) = x^{-1} \]

### Step 6: Integrate both sides
Integrating both sides gives:
\[ y \cdot x^{\frac{1}{2} \log x} = \int x^{-1} \, dx = \log x + C \]

### Step 7: Solve for \( y \)
Now, we can solve for \( y \):
\[ y = \frac{\log x + C}{x^{\frac{1}{2} \log x}} \]

### Step 8: Final form of the solution
Rearranging gives us:
\[ y = x^{1 - \frac{1}{2} \log x} + C x^{-\frac{1}{2} \log x} \]

### Conclusion
Thus, the general solution of the differential equation is:
\[ y = x^{1 - \frac{1}{2} \log x} + C x^{-\frac{1}{2} \log x} \]