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Question Number 182039 by liuxinnan last updated on 03/Dec/22

i f f^{‴} (x) = f (x)

f i n d w h a t i s f (x)

Answered by FelipeLz last updated on 03/Dec/22

f^{‴} (x) - f (x) = 0

f (x) = e^{r x} \to r^{3} e^{r x} - e^{r x} = 0

e^{r x} (r^{3} - 1) = 0

e^{r x} \neq 0 \forall x \to r^{3} - 1 = 0

(r - 1) (r^{2} + r + 1) = 0

r_{1} = 1, r_{2} = - \frac{1}{2} (1 - \sqrt{3} i), r_{3} = - \frac{1}{2} (1 + \sqrt{3} i)

f (x) = c_{1} e^{x} + c_{2} \frac{1}{\sqrt{e^{x}}} \cos (\frac{\sqrt{3}}{2} x) + c_{3} \frac{1}{\sqrt{e^{x}}} \sin (\frac{\sqrt{3}}{2} x)

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