Question

# (2y+xy)dx+2xdy=0

Second order linear equations
$$\displaystyle{\left({2}{y}+{x}{y}\right)}{\left.{d}{x}\right.}+{2}{x}{\left.{d}{y}\right.}={0}$$

2020-12-17

Here the Differential equations is given by
$$\displaystyle{\left({2}{y}+{x}{y}\right)}{\left.{d}{x}\right.}+{2}{x}{\left.{d}{y}\right.}={0}.$$
This can be written as
$$y(2+x)dx+2xdy=0 \Rightarrow \frac{2+x}{x}dx+\frac{2}{y}dy=0$$
$$\Rightarrow (\frac{2}{x}+1)dx+\frac{2}{y}dy=0$$

$$\Rightarrow d(2\ln x+x)+d(2\ln y)=0$$

$$\Rightarrow d(2\ln x+x+2\ln y)=0$$

$$\Rightarrow ∫d(2\ln x+x+2\ln y)=c$$

$$\Rightarrow 2\ln x+x+2\ln y=c$$

$$\Rightarrow \ln y=(c-2\ln x-x)/2$$

$$\Rightarrow y=e^{\frac{c-2\ln x-x}{2}}$$
where cc is a arbitrary constants.