Let y′′′−9y′′+20y′=0. find all values of r such that y=erx satisfies the differential equation. if there is more than one correct answer, enter your answers as a comma separated list.

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Respuesta :

rejkjavik rejkjavik · Answer 1 · 2018-09-07T22:42:24+02:00

we are given

differential equation as

[tex]y'''-9y''+20y'=0[/tex]

we are given

[tex]y=e^{rx}[/tex]

Firstly, we will find y' , y'' and y'''

those are first , second and third derivative

First derivative is

[tex]y'=re^{rx}[/tex]

Second derivative is

[tex]y''=r*re^{rx}[/tex]

[tex]y''=r^2e^{rx}[/tex]

Third derivative is

[tex]y'''=r^2*re^{rx}[/tex]

[tex]y'''=r^3e^{rx}[/tex]

now, we can plug these values into differential equation

and we get

[tex]r^3 e^{rx}-9r^2 e^{rx}+20re^{rx}=0[/tex]

now, we can factor out common terms

[tex]e^{rx}(r^3 -9r^2 +20r)=0[/tex]

we can move that term on right side

[tex](r^3 -9r^2 +20r)=0[/tex]

now, we can factor out

[tex]r(r^2 -9r +20)=0[/tex]

[tex]r(r-5)(r-4)=0[/tex]

now, we can set them equal

[tex]r=0[/tex]

[tex]r-5=0[/tex]

[tex]r=5[/tex]

[tex]r-4=0[/tex]

[tex]r=4[/tex]

so, we will get

[tex]r=0,4,5[/tex]...............Answer