Q.Solve the differential equation $x\,dy + y\,dx = 0$ (for $x, y \neq 0$), expressing the solution as a relation between $x$ and $y$.

Divide through by $xy$ to separate the variables, then integrate.

Starting from $x\,dy + y\,dx = 0$, first isolate $x\,dy$ on one side: $x\,dy = -y\,dx$.

Dividing both sides by $xy$ to separate the variables: $\dfrac{dy}{y} = -\dfrac{dx}{x}$.

Integrating both sides: $\ln|y| = -\ln|x| + c$, i.e. $\ln|y| + \ln|x| = c$, so $\ln|xy| = c$.

Exponentiating: $|xy| = e^c$, and writing $k=\pm e^c$ (an arbitrary nonzero constant absorbing the sign), the solution simplifies to $xy = k$. …

Alternative — recognising $x\,dy+y\,dx$ directly as $d(xy)$ (the product rule written in differential form): the equation $d(xy)=0$ integrates immediately …

A frequent error is dividing by $xy$ incorrectly, e.g. writing $\dfrac{dy}{x}=-\dfrac{dx}{y}$ (dividing $dy$ by the wrong variable's factor) instead of the correct $\dfrac{dy}{y}=-\dfrac{dx}{x}$ — each differential must be div …