Xy’-y=2sqrt(xy)

To solve the given equation \(xy' - y = 2\sqrt{xy}\), we can follow these steps:

Let's denote \(y'\) as the derivative of \(y\) with respect to \(x\).

1. Rewrite the equation: \[ xy' - y = 2\sqrt{xy} \]

2. Divide both sides by \(x\): \[ y' - \frac{y}{x} = 2\sqrt{\frac{y}{x}} \]

3. Let \(u = \frac{y}{x}\). Then, we have: \[ y' = u'x +u \]

4. Substitute \(u = \frac{y}{x}\) and \(y' = u'x +u\) into the equation: \[ u'x + u - u = 2\sqrt{u} \] \[ u'x = 2\sqrt{u} \]

5. Divide both sides by \(2\sqrt{u}\) and \(x\): \[ \frac{u'}{2\sqrt{u}} = \frac{1}{x} \]

6. Integrate both sides with respect to \(x\): \[ \int \frac{1}{2\sqrt{u}} du = \int \frac{1}{x} dx \] \[ \sqrt{u} = \ln|x| + C \]

7. Substitute back \(u = \frac{y}{x}\): \[ \sqrt{\frac{y}{x}} = \ln|x| + C \]

This is the general solution to the differential equation \(xy' - y = 2\sqrt{xy}\).