The solution of the differential equation `(dy)/(dx) = (y^(2))/(xy-x^(2))` is

Options

A. `e^(y//x)=k x`
B. `e^(y//x)=k y`
C. `e^(x//y)= k x`
D. `e^(-y//x)=k y`

Explanation

To solve the differential equation \(\frac{dy}{dx} = \frac{y^2}{xy - x^2}\), we will use the substitution \(y = vx\), where \(v\) is a function of \(x\). Let's go through the steps one by one.

### Step 1: Substitute \(y = vx\)
We start by substituting \(y\) with \(vx\):
\[ \frac{dy}{dx} = v + x\frac{dv}{dx} \]
Now, substituting \(y\) in the differential equation:
\[ \frac{dy}{dx} = \frac{(vx)^2}{x(vx) - x^2} = \frac{v^2x^2}{x^2(v - 1)} = \frac{v^2}{v - 1} \]

### Step 2: Set the equations equal
Now we equate the two expressions we have for \(\frac{dy}{dx}\):
\[ v + x\frac{dv}{dx} = \frac{v^2}{v - 1} \]

### Step 3: Rearranging the equation
Rearranging gives:
\[ x\frac{dv}{dx} = \frac{v^2}{v - 1} - v \]
Simplifying the right side:
\[ x\frac{dv}{dx} = \frac{v^2 - v(v - 1)}{v - 1} = \frac{v^2 - v^2 + v}{v - 1} = \frac{v}{v - 1} \]

### Step 4: Separate variables
Now we can separate the variables:
\[ \frac{v - 1}{v} dv = \frac{1}{x} dx \]

### Step 5: Integrate both sides
Integrating both sides:
\[ \int \left(1 - \frac{1}{v}\right) dv = \int \frac{1}{x} dx \]
This gives:
\[ v - \log |v| = \log |x| + C \]

### Step 6: Substitute back for \(y\)
Recall that \(v = \frac{y}{x}\), substituting back gives:
\[ \frac{y}{x} - \log \left|\frac{y}{x}\right| = \log |x| + C \]
Multiplying through by \(x\):
\[ y - x \log |y| + x \log |x| = Cx \]

### Step 7: Final form
Rearranging gives:
\[ y - x \log |y| = Cx - x \log |x| \]
This is the implicit solution of the differential equation.