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Question Number 183211 by Mastermind last updated on 23/Dec/22

y^((iv)) +16y^((iii)) +9y^((ii)) +256y^((i)) +256y=0      M.m

$$\mathrm{y}^{\left(\mathrm{iv}\right)} +\mathrm{16y}^{\left(\mathrm{iii}\right)} +\mathrm{9y}^{\left(\mathrm{ii}\right)} +\mathrm{256y}^{\left(\mathrm{i}\right)} +\mathrm{256y}=\mathrm{0} \\ $$$$ \\ $$$$ \\ $$$$\mathrm{M}.\mathrm{m} \\ $$

Answered by aleks041103 last updated on 23/Dec/22

y=e^(rx)   ⇒r^4 +16r^3 +9r^2 +256r+256=0  r=r_(1,2,3,4)   ⇒y(x)=Ae^(r_1 x) +Be^(r_2 x) +Ce^(r_3 x) +De^(r_4 x)

$${y}={e}^{{rx}} \\ $$$$\Rightarrow{r}^{\mathrm{4}} +\mathrm{16}{r}^{\mathrm{3}} +\mathrm{9}{r}^{\mathrm{2}} +\mathrm{256}{r}+\mathrm{256}=\mathrm{0} \\ $$$${r}={r}_{\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4}} \\ $$$$\Rightarrow{y}\left({x}\right)={Ae}^{{r}_{\mathrm{1}} {x}} +{Be}^{{r}_{\mathrm{2}} {x}} +{Ce}^{{r}_{\mathrm{3}} {x}} +{De}^{{r}_{\mathrm{4}} {x}} \\ $$

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