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

Question
Differential equations
asked 2020-12-16
\(\displaystyle{\left({2}{y}+{x}{y}\right)}{\left.{d}{x}\right.}+{2}{x}{\left.{d}{y}\right.}={0}\)

Answers (1)

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
PSKy(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}}ZSK
where cc is a arbitrary constants.
0

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