2nd order auxillary differentials Watch

IceKidd
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A couple of issues withbdifferential equations

1.When the auxillary equation has repeated roots, you are left with one complimentary function so one constant of integration which doesnt follow linearly as it is 2nd order. So x (Ae^nx) is another standard one, which the book im using shows its true. But it doesnt explain why, can anyone here

2. For non homogeneous functions when you are finding the particular integral, you use a trial function that is of same form as RHS of differential. It does work obviously and I kinf of get why, but any explicit reasoning.

3.Also where the RHS is of the same form as the complimentary function the trial function is x times the trial function u would normally use for the RHS. I can see that using the trial function of same form of RHS insnt going to lead anywhere in thus situation. But why is it x timesd by it that works, why not something completely different or x^2 or something?
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brianeverit
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(Original post by IceKidd)
A couple of issues withbdifferential equations

1.When the auxillary equation has repeated roots, you are left with one complimentary function so one constant of integration which doesnt follow linearly as it is 2nd order. So x (Ae^nx) is another standard one, which the book im using shows its true. But it doesnt explain why, can anyone here

2. For non homogeneous functions when you are finding the particular integral, you use a trial function that is of same form as RHS of differential. It does work obviously and I kinf of get why, but any explicit reasoning.

3.Also where the RHS is of the same form as the complimentary function the trial function is x times the trial function u would normally use for the RHS. I can see that using the trial function of same form of RHS insnt going to lead anywhere in thus situation. But why is it x timesd by it that works, why not something completely different or x^2 or something?
I have attached a course in differential equations that I wrote for the MEI module, Check out Lessons 5,6 and 7 and they should answer your queries.
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