# Find transient terms in this general solution to a differential equation, if there are any y=(x+C)(frac{x+2}{x-2})

Find transient terms in this general solution to a differential equation, if there are any
$$\displaystyle{y}={\left({x}+{C}\right)}{\left({\frac{{{x}+{2}}}{{{x}-{2}}}}\right)}$$

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unett

$$\displaystyle{\left({x}+{C}\right)}{\left({\frac{{{x}+{2}}}{{{x}-{2}}}}\right)}={\frac{{{x}^{{2}}}}{{{x}-{2}}}}+{\frac{{{\left({2}+{C}\right)}{x}}}{{{x}-{2}}}}+{\frac{{{2}{C}}}{{{x}-{2}}}}$$
Then take the limits of each of these three terms as $$x\rightarrow\infty$$ and see which(if any) approach 0. All three are rational expressions, so it is clear that only $$\displaystyle{\frac{{{2}{C}}}{{{x}-{2}}}}$$ is transient term.