Step-by-Step Solution

The given differential equation is: \(x \, dy - y \, dx - \sqrt{x^{2} + y^{2}} \, dx = 0\)

Step 1: Rearrange the equation

Rearrange the terms to isolate \(dy/dx\):

\(x \, dy = (y + \sqrt{x^{2} + y^{2}}) \, dx\)

\(\frac{dy}{dx} = \frac{y}{x} + \frac{\sqrt{x^{2} + y^{2}}}{x}\)

\(\frac{dy}{dx} = \frac{y}{x} + \sqrt{1 + \frac{y^{2}}{x^{2}}}\)

Step 2: Recognize the homogeneous form

The equation is homogeneous. Let \(y = vx\), so \(\frac{dy}{dx} = v + x\frac{dv}{dx}\)

Step 3: Substitute and simplify

Substitute \(y = vx\) into the equation:

\(v + x\frac{dv}{dx} = v + \sqrt{1 + v^{2}}\)

\(x\frac{dv}{dx} = \sqrt{1 + v^{2}}\)

Step 4: Separate variables and integrate

Separate the variables:

\(\frac{dv}{\sqrt{1 + v^{2}}} = \frac{dx}{x}\)

Integrate both sides:

\(\int \frac{dv}{\sqrt{1 + v^{2}}} = \int \frac{dx}{x}\)

\(\sinh^{-1}(v) = \ln|x| + C\)

Step 5: Substitute back \(v = y/x\)

\(\sinh^{-1}\left(\frac{y}{x}\right) = \ln|x| + C\)

Step 6: Express in exponential form

Taking sinh of both sides:

\(\frac{y}{x} = \sinh(\ln|x| + C)\)

\(y = x \sinh(\ln|x| + C)\)

Alternatively, using the logarithmic form of inverse hyperbolic sine:

\(\ln\left(\frac{y}{x} + \sqrt{1 + \frac{y^2}{x^2}}\right) = \ln|x| + C\)

\(\frac{y}{x} + \sqrt{1 + \frac{y^2}{x^2}} = e^{\ln|x| + C} = e^C |x| = Kx\), where \(K = e^C\)

\(y + \sqrt{x^2 + y^2} = Kx^2\)

\(\sqrt{x^2 + y^2} = Kx^2 - y\)

\(x^2 + y^2 = (Kx^2 - y)^2 = K^2x^4 - 2Kx^2y + y^2\)

\(x^2 = K^2x^4 - 2Kx^2y\)

\(1 = K^2x^2 - 2Ky\)

\(2Ky = K^2x^2 - 1\)

\(y = \frac{K^2x^2 - 1}{2K}\)

\(y = Ax^2 + B\), where \(A = K/2\) and \(B = -1/(2K)\)