What is factor e^{-at}u(t)+e^{at}u(-t) ?

The solution to factor e^{-at}u(t)+e^{at}u(-t) is -e^{-at}tu(e^{at}+1)(e^{at}-1)

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factor e^{-at}u(t)+e^{at}u(-t)

factor

e^{- a t} u (t) + e^{a t} u (- t)

- e^{- a t} t u (e^{a t} + 1) (e^{a t} - 1)

Solution steps

e^{- a t} u (t) + e^{a t} u (- t)

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

e^{a t} = e^{- a t} e^{2 a t}

= e^{- a t} u t + e^{- a t} e^{2 a t} u (- t)

Factor out common term

t u e^{- a t}

= t u e^{- a t} (1 - e^{2 a t})

Factor

- e^{2 a t} + 1 : - (e^{a t} + 1) (e^{a t} - 1)

= e^{- a t} t u (- (e^{a t} + 1) (e^{a t} - 1))

Refine

= - e^{- a t} t u (e^{a t} + 1) (e^{a t} - 1)

A (BC) + C D

- 5 \cdot 0

cos (x^{2} + x)

sin (n) (- π)

(x \cdot a) (y \cdot b)

What is factor e^{-at}u(t)+e^{at}u(-t) ?
The solution to factor e^{-at}u(t)+e^{at}u(-t) is -e^{-at}tu(e^{at}+1)(e^{at}-1)