Problem Statement
Solve the differential equation $(x^2 - y^2)dx + 2xy\,dy = 0$, given that $y = 1$ when $x = 1$.
Verified Solution & Marking Scheme
Rearrange as dy/dx
$$\frac{dy}{dx} = \frac{y^2 - x^2}{2xy}$$
Substitute y = vx
$$v + x\frac{dv}{dx} = \frac{v^2 - 1}{2v} \implies x\frac{dv}{dx} = -\frac{v^2 + 1}{2v}$$
Separate and Integrate
$$\int \frac{2v}{v^2 + 1} dv = -\int \frac{dx}{x} \implies \ln(v^2 + 1) = -\ln|x| + \ln C \implies x^2 + y^2 = Cx$$
Apply Condition y(1) = 1
$$1 + 1 = C(1) \implies C = 2 \implies x^2 + y^2 = 2x$$