Show that this is a homogeneous differential equation.

Find the general solution, giving your answer in the form $y = f(x)$ .

the equation can be rewritten as

$\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{y + \sqrt {{x^2} - {y^2}} }}{x} = \frac{y}{x} + \sqrt {1 - {{\left( {\frac{y}{x}} \right)}^2}}$     A1

so the differential equation is homogeneous     AG

[1 mark]

put y = vx so that $\frac{{{\text{d}}y}}{{{\text{d}}x}} = v + x\frac{{{\text{d}}v}}{{{\text{d}}x}}$     M1A1

substituting,

$v + x\frac{{{\text{d}}v}}{{{\text{d}}x}} = v + \sqrt {1 - {v^2}}$     M1

$\int {\frac{{{\text{d}}v}}{{\sqrt {1 - {v^2}} }} = \int {\frac{{{\text{d}}x}}{x}} }$     M1

$\arcsin v = \ln x + C$     A1

$\frac{y}{x} = \sin (\ln x + C)$     A1

$y = x\sin (\ln x + C)$     A1

[7 marks]