Question

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

Second order linear equations
ANSWERED
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
\(y(2+x)dx+2xdy=0  \Rightarrow  \frac{2+x}{x}dx+\frac{2}{y}dy=0\)
\( \Rightarrow  (\frac{2}{x}+1)dx+\frac{2}{y}dy=0\)

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

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

\(\Rightarrow  ∫d(2\ln x+x+2\ln y)=c\)

\(\Rightarrow  2\ln x+x+2\ln y=c\)

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

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

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