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Differential Equations

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An Application of Partial Fraction Decomposition

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Consider $f(t)=\cfrac{e^{3t}-e^{2t}+1}{e^{4t}-e^{2t}-2}$. This function is not a rational function.

However, we can simplify it using the techniques from the partial fraction decomposition.

The technique is to regard $e^t$ as a variable. Which of the following equals $f(t)$?

A

$\cfrac{(2/3)e^{2t}-1}{e^{2t}+2}+\cfrac{1}{6(e^t+1)}+ \cfrac{1}{6(e^t-1)}$

B

$\cfrac{e^t-2}{3(e^{2t}+1)}+\cfrac{4+\sqrt{2}}{12}\cfrac{1}{e^t+\sqrt{2}} +\cfrac{4-\sqrt{2}}{12}\cfrac{1}{e^t-\sqrt{2}}$

C

$\cfrac{e^{2t}-5/3}{e^{2t}+2}+\cfrac{1}{6(e^t+1)}- \cfrac{1}{6(e^t-1)}$

D

$\cfrac{e^t-2}{3(e^{2t}+1)}+\cfrac{2\sqrt{2}-1}{6\sqrt{2}}\frac{1}{e^t+\sqrt{2}} +\cfrac{2\sqrt{2}+1}{6\sqrt{2}}\frac{1}{e^t-\sqrt{2}}$