Solution of the differential equation `ydx-x dy+logx dx=0` is

Options

A. `y+log|x|+1=cx`
B. `y-log|x|+1=cx`
C. `y+log|x|-1=cx`
D. `y-log|x|-1=cx`

Answer

To solve the differential equation \( y \, dx - x \, dy + \log x \, dx = 0 \), we can follow these steps:

### Step 1: Rearranging the Equation
We start with the given equation:
\[ y \, dx - x \, dy + \log x \, dx = 0 \]
We can rearrange this to group the \( dx \) terms:
\[ (y + \log x) \, dx = x \, dy \]

### Step 2: Dividing by \( dx \)
Next, we divide both sides by \( dx \) (assuming \( dx \neq 0 \)):
\[ y + \log x = x \frac{dy}{dx} \]
Rearranging gives us:
\[ \frac{dy}{dx} = \frac{y + \log x}{x} \]

### Step 3: Identifying the Form
This equation can be rewritten in the standard linear form:
\[ \frac{dy}{dx} - \frac{y}{x} = \frac{\log x}{x} \]
Here, we identify \( p = -\frac{1}{x} \) and \( q = \frac{\log x}{x} \).

### Step 4: Finding the Integrating Factor
The integrating factor \( I.F. \) is given by:
\[ I.F. = e^{\int p \, dx} = e^{\int -\frac{1}{x} \, dx} = e^{-\log x} = \frac{1}{x} \]

### Step 5: Multiplying the Equation by the Integrating Factor
We multiply the entire differential equation by the integrating factor:
\[ \frac{1}{x} \frac{dy}{dx} - \frac{y}{x^2} = \frac{\log x}{x^2} \]

### Step 6: Integrating Both Sides
Now we can integrate both sides:
\[ \int \left( \frac{1}{x} \frac{dy}{dx} - \frac{y}{x^2} \right) dx = \int \frac{\log x}{x^2} \, dx \]
The left-hand side simplifies to:
\[ \frac{y}{x} \]
For the right-hand side, we will use integration by parts, letting:
- \( u = \log x \) and \( dv = \frac{1}{x^2} dx \)
- Then, \( du = \frac{1}{x} dx \) and \( v = -\frac{1}{x} \)

Using integration by parts:
\[ \int u \, dv = uv - \int v \, du \]
We have:
\[ \int \log x \cdot \frac{1}{x^2} \, dx = -\frac{\log x}{x} - \int -\frac{1}{x^2} \, dx = -\frac{\log x}{x} + \frac{1}{x} \]

### Step 7: Putting it All Together
Now we can write:
\[ \frac{y}{x} = -\frac{\log x}{x} + \frac{1}{x} + C \]
Multiplying through by \( x \):
\[ y = -\log x + 1 + Cx \]

### Final Step: Writing the General Solution
Thus, the general solution of the differential equation is:
\[ y + \log |x| + 1 = Cx \]