42. Integrate the differential equation

$$
\left(1+x^{2}\right) d y-2 x y d x=0
$$
Solution. The given equation is an equation with separable variables. By dividing both sides of the equation by the product $y\left(1+x^{2}\right)$, we obtain an equation with separated variables:

$$
\frac{d y}{y}-\frac{2 x d x}{1+x^{2}}=0
$$

Integrating this equation, we find

$$
\ln |y|-\ln \left(1+x^{2}\right)=\ln |C| \text { or } \ln \left(\frac{|y|}{1+x^{2}}\right)=\ln |C|
$$

from which we obtain the general solution $y=C\left(1+x^{2}\right)$.

Remark. When dividing by $y\left(1+x^{2}\right)$, it was assumed that $y\left(1+x^{2}\right) \neq 0$, i.e., $y \neq 0,1+x^{2} \neq 0$. However, $y=0$ is a solution of the equation, which can be verified directly. This solution is obtained from the general solution when $C=0$.

Based on the solved examples, the algorithm for solving a differential equation with separable variables is evident.

$1^{0}$. Express the derivative of the function through differentials $d x$ and $d y$.

$2^{0}$. Terms with the same differentials are moved to one side of the equality and the differential is factored out.

$3^{0}$. Separate the variables.

$4^{0}$. Integrate both sides of the equality and find the general solution.

$5^{0}$. If initial conditions are given, find the particular solution.

Depending on the form of the equation, some steps of the solution algorithm may be omitted.