General solution of `y -x (dy)/(dx) = 0`

Options

A. 3 log x `+ (7)/(y) = c`
B. 2 log x `+ (3)/(y) = c`
C. log x - log y = log c
D. 3 log x `+ (2)/(x) = c`

Answer

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

### Step 1: Rearrange the equation
We start with the given equation:
\[ y - x \frac{dy}{dx} = 0 \]
Rearranging gives:
\[ x \frac{dy}{dx} = y \]

### Step 2: Separate variables
Next, we can separate the variables \( y \) and \( x \):
\[ \frac{dy}{y} = \frac{dx}{x} \]

### Step 3: Integrate both sides
Now we integrate both sides:
\[ \int \frac{dy}{y} = \int \frac{dx}{x} \]
This results in:
\[ \ln |y| = \ln |x| + C \]
where \( C \) is the constant of integration.

### Step 4: Exponentiate to eliminate the logarithm
To eliminate the logarithm, we exponentiate both sides:
\[ |y| = e^{\ln |x| + C} = |x| e^C \]
Let \( k = e^C \) (where \( k \) is a positive constant), we can write:
\[ y = kx \]

### Step 5: General solution
Thus, the general solution of the differential equation is:
\[ y = Cx \]
where \( C \) is a constant.