# Show that y= cx - 2 is a general solution of xy'=y+ 2, where c is an arbitrary constant and find c if y(1)=3

Show that y= cx - 2 is a general solution of xy'=y+ 2, where c isan arbitrary constant and find c if y(1)=3
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Shannon Andrews
The differential equation can be separated: dy/(y+2)=dx/x.Integration: $\mathrm{ln}\left(y+2\right)=\mathrm{ln}\left(cx\right)$. Exponentiated: y+2=cx. So y=cx-2 is shown. $\int dx/x=\mathrm{ln}\left(x\right)+C=\mathrm{ln}\left(x\right)+\mathrm{ln}\left(\mathrm{exp}\left(C\right)\right)=\mathrm{ln}\left(x\mathrm{exp}\left(C\right)\right)=\mathrm{ln}\left(cx\right)$.y(1)=3=3c-2 is solved by c=5/3. y(x)=5x/3-2.