Differential equations with separable variables and those reducible to them.
Equations with separable variables.
Equations with separable variables can be written in the form$$y'=f(x)g(y),\qquad\qquad\qquad(1)$$ and also in the form $$M(x)N(y)dx+P(x)Q(y)dy=0.\qquad\qquad\qquad(2)$$ o solve this equation, it needs to be transformed in such a way that one part of the equation involves only $x$, and the other only involves $y$, and then integrate both parts.
When dividing both sides of the equation by an expression containing the unknowns $x$ and $y$, solutions that make this expression zero may be lost.
Example. Solve the equation $$x^2y^2y'+1=y.$$
Solution.
Let's transform the given equation into the form (2).
$$x^2y^2\frac{dx}{dy}=y-1;\qquad\qquad x^2y^2dy=(y-1)dx.$$
We divide both sides of the equation by $x^2(y-1):$
$$\frac{y^2}{y-1}dy=\frac{dx}{x^2}.$$
The variables are separated. We integrate both sides of the equation:
$$\int\frac{y^2}{y-1}dy=\int\frac{dx}{x^2}.$$
$$\int\frac{y^2}{y-1}dy=\int\frac{y^2-1+1}{y-1}dy=\int\left(y+1+\frac{1}{y-1}\right)dy=\frac{y^2}{2}+y+\ln|y-1|+C.$$
$$\int\frac{dx}{x^2}=-\frac{1}{x}+C.$$
Thus,
$$\frac{y^2}{2}+y+\ln|y-1|=-\frac{1}{x}+C.$$
When dividing by $x^2(y-1)$, solutions $x=0$ and $y-1=0,$ i.e., $y=1,$ could have been lost. If we substitute these values into the original equation, it becomes evident that $y=1$ is a solution to the given equation, whereas $x=0$ is not.
Answer: $\frac{y^2}{2}+y+\ln|y-1|=-\frac{1}{x}+C,$ $y=1.$
Equations reducible to equations with separable variables.
Equations of the form $y'=f(ax+by)$ are reduced to equations with separable variables by the substitution $z=ax+by$ (or $z=ax+by+c,$ where $c$ is any constant).
Examples.
1. $y'=\sqrt{4x+2y-1}.$
Solution.
Let's make the substitution.
$z=4x+2y-1.$ From this, $y=\frac{1}{2}(z-4x+1)\Rightarrow y'=\frac{1}{2}z'-2$
We obtain$$\frac{1}{2}z'-2=\sqrt{z}$$
Let's transform this equation into the form (2).
$$\frac{dz}{2dx}=\sqrt{z}+2\Rightarrow \frac{dz}{\sqrt z+2}=2dx$$
We integrate both sides of the equation:
$$\int\frac{dz}{\sqrt z+2}=\int 2dx$$
$$\int\frac{dz}{\sqrt z+2}=[z=t^2\qquad dz=2tdt]=\int\frac{2tdt}{t+2}=2\int\frac{t+2-2}{t+2}dt=2(t-2\ln|t+2|)+C=$$ $$=2(\sqrt z-2\ln|\sqrt z+2|)+C=2(\sqrt{4x+2y-1}-2\ln|\sqrt{4x+2y-1}+2|)+C.$$
$$\int 2dx=2x+C.$$
Thus, we obtained:
$$2(\sqrt{4x+2y-1}-2\ln|\sqrt{4x+2y-1}+2|)=2x+C\Rightarrow$$ $$\Rightarrow\sqrt{4x+2y-1}-\ln(\sqrt{4x+2y-1}+2)=x+C.$$
Answer: $$\sqrt{4x+2y-1}-\ln(\sqrt{4x+2y-1}+2)=x+C..$$
