Method of undetermined coefficients guess for particular solution

[neat ridge]

hi! i was doing this problem but i'm not too sure why the particular solution for this ODE includes a second factor of t.

[plain fern]

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[vital hamlet]

Have you found the roots of the characteristic equation?

[neat ridge]

im not sure if im misunderstanding smth but based on tables like these:

i assumed the form of the particular solution would be the other two options

[neat ridge]

yep!

3 and -5

[vital hamlet]

Right.
And what root does the quasipolynomial on the RHS correspond to?

[neat ridge]

ohh

the first one

[vital hamlet]

Yeah! It corresponds to 3, which is a root of our CE with multiplicity 1.

So, we also multiply the particular solution by t.

[neat ridge]

does this have smth to do with it not being linearly independent to one of the homogeneous solutions..? i remember learning smth along those lines in class

[neat ridge]

ahh i see

[vital hamlet]

Yeah, something like that.

[neat ridge]

thank you so much!!

[vital hamlet]

You're welcome!

[neat ridge]

+close