Question about solving a differential equation D(D-3)(D+4)[y]=0, where D is the differential

burkinaval1b

burkinaval1b

Answered question

2022-01-20

Question about solving a differential equation
D(D3)(D+4)[y]=0,
where D is the differential operator, how to get the general solution of y? The solution suggest that it is
y=6c12c2exp(4t)+3c3exp(3t)

Answer & Explanation

scomparve5j

scomparve5j

Beginner2022-01-20Added 38 answers

Given the arbitrary 6, -2 and 3 in front of the constants, the exercise might want you to solve three successive differential equations -first Du=0,then (D3)v=u and finally (D+4)w=v.
This process can be circumnavigated with a few observations. First, the differential equation is third-order and thus has three linearly independent solutions. Second, the three operators D,D3 and D+4 all commute. Third, D+a annihilates 0 for any constant a (that is, D+a applied to the zero function returns the zero function). Taken together, this implies that any solution to one of
{(D+0)ϕ=0(i)(D3)ϕ=0(ii)(D+4)ϕ=0(iii)
is also a solution to the original differential equation. To see this for e.g. (i), take a solution to the differential equation Dϕ=0 and observe
D(D3)(D+4)ϕ=(D3)(D+4)Dϕ=(D3)(D+4)0=(D3)0=0,
Finally, find the general solutions to (i), (ii) and (iii) and observe they are all linearly independent, and thus form a basis for the solution space we are after.
Orlando Paz

Orlando Paz

Beginner2022-01-21Added 42 answers

Heres
RizerMix

RizerMix

Expert2022-01-27Added 656 answers

I'm assuming that this is for an introductory mathematical physics course or something like that. Anyways. Just do it one at a time. I would suggest writing (D3)(D+4)g(x)=1D0, then conintinuing.

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