27. Solve the equation $x d x+y d y=0$.
Solution. Here the variables are separated. Integrating, we get

$$
\int x d x+\int y d y=C ; \frac{x^{2}}{2}+\frac{y^{2}}{2}=C ; x^{2}+y^{2}=2 C .
$$

Since $C$ is arbitrary, we can denote $2 C$ by $C^{2}$, considering that the left-hand side of the last equation is positive. Then this equation will take the form $x^{2}+y^{2}=C^{2}$. This is the general solution, or as they say, the general integral of the given differential equation.

From a geometric point of view, we have obtained a family (set) of concentric circles centered at the origin with a radius equal to $C$ (compare the obtained equation with the known equation of a circle of the form $x^{2}+y^{2}=R^{2}$).