Question Number 182039 by liuxinnan last updated on 03/Dec/22
$${if}\:{f}'''\left({x}\right)={f}\left({x}\right) \\ $$$${find}\:{what}\:{is}\:{f}\left({x}\right)\: \\ $$
Answered by FelipeLz last updated on 03/Dec/22
$${f}'''\left({x}\right)−{f}\left({x}\right)\:=\:\mathrm{0} \\ $$$${f}\left({x}\right)\:=\:{e}^{{rx}} \:\rightarrow\:{r}^{\mathrm{3}} {e}^{{rx}} −{e}^{{rx}} \:=\:\mathrm{0} \\ $$$${e}^{{rx}} \left({r}^{\mathrm{3}} −\mathrm{1}\right)\:=\:\mathrm{0} \\ $$$${e}^{{rx}} \:\neq\:\mathrm{0}\:\forall{x}\:\rightarrow\:{r}^{\mathrm{3}} −\mathrm{1}\:=\:\mathrm{0} \\ $$$$\left({r}−\mathrm{1}\right)\left({r}^{\mathrm{2}} +{r}+\mathrm{1}\right)\:=\:\mathrm{0} \\ $$$${r}_{\mathrm{1}} \:=\:\mathrm{1},\:{r}_{\mathrm{2}} \:=\:−\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}−\sqrt{\mathrm{3}}{i}\right),\:{r}_{\mathrm{3}} \:=\:−\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}+\sqrt{\mathrm{3}}{i}\right) \\ $$$${f}\left({x}\right)\:=\:{c}_{\mathrm{1}} {e}^{{x}} +{c}_{\mathrm{2}} \frac{\mathrm{1}}{\:\sqrt{{e}^{{x}} }}\mathrm{cos}\left(\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}{x}\right)+{c}_{\mathrm{3}} \frac{\mathrm{1}}{\:\sqrt{{e}^{{x}} }}\mathrm{sin}\left(\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}{x}\right) \\ $$