DONE! - Yet ANOTHER Linear Algebra problem...

[ChineseGuy]

Ok, so here is the problem:

"Find 4 particular solutions of D^4 y = 0 with state vectors k!Ek for k=1,2,3,4."

So the problem gave me the 4th differential of y = zero, hmm.. ok

and state vectors [1,0,0,0]T [0,2,0,0]T [0,0,6,0]T [0,0,0,24]T

What do i do with the D^4? usually the problems I have seen its (D+1) or something, where there is an eigenvalue for me to just stick it in the homogeneous equation. in this example it would have been like yh=C1*e^-t
But since the eigenvalues are all zeros, repeated 4 times, does that mean my yh=c1+c2+c3+c4? That doesnt make any sense, how would i use that with the statevectors to find 4 particular solutions (yp)? Any help would be greatly appreciated.

 

[ChineseGuy]

Well, after dinner, I sat down and looked at it some more, and it just popped out at me, you guys tell me if this makes any sense.

1) We know that the 4th derivative is zero

2) From statement 1, we can deduce that the 3rd derivative is a constant. (C3)

3) From the statevectors given, if this it goes y, y', y" , y''' ... then the initial state that correspond to the 3rd derivative is 24

4) From statement 2, we can deduce that y'''=24

5) Take the antiderivative of y''', to get to y'', just do chain rule, and y" = 24x + C2

6) since we are given state vectors, those are initial conditions, y''(0)=6=24(0)+C2; we now know that C2 is 6

7) antiderivative of that, y'(x)=12x^2+6x+C1 ; plug in the initial conditions, C1 = 2

8) antiderivative of that. y(x)=4x^3+3x^2+2x+C ; plug in the initial conditions C0=1

9) soo following those statements, y(x)=4x^3+3x^2+2x+1

That's all I got so far. What do you guys think?

 

[esun]

Hmmm, sorry, I don't know how to do this problem. I don't recall learning about "state vectors" and I don't understand what k!Ek means.

 

[ChineseGuy]

k! is k factorial... from 1 to 4 is what's given..

Ek... that is just [k,0,0,0....]T [0,k,0,0...]T to whatever your dimension is.

so i did that [1,0,0,0]T [0,2!,0,0]T [0,0,3!,0]T [0,0,0,4!]T ... which turned out to be [1,0,0,0]T [0,2,0,0]T [0,0,6,0]T [0,0,0,24]T
state vectors are initial values, constrains that allow you to find your constants to allow you to find your particular solution.

:-\ Think I might have gotten this one. the y''',y'',y',y would be the 4 particular solutions my prof is asking I am assuming, since all the constants (c0,c1,c2,c3) are defined.

 

[CSMR]

What you have said is not complete nonsense, something can be salvaged, but you are extremely confused about what you are doing.

You have an "answer". What is the "state vector" corresponding to your answer? Or are there several "state vectors" associated with your answer?

 

[CycloWizard]

The general solution of D^4(y)=0 is found by simply integrating four times and is y= c3*x^3+c2*x^2+c1*x+c0. I'm not sure what eigenvalues have to do with it. I also do not understand how the state vectors are defined. What is E? Is it k!*E*k?

 

[ChineseGuy]

E is the greek letter epsilon, my professor never really did defined it, he just gave us examples of it.. lets see if I can get the symbol to come up right...

k! ek (the second k is suppose to be a subscript) .... soo k factorial multiplied by the "epsilon sub 1,2..." soo it would be like k!*[1,0,0,0]T k!*[0,1,0,0]T ... where the 1 moves down to the next row... I dont know if I am making a whole lot of sense. Oh well, I just turned the assignment in, so I dont have it in front of me anymore. At this point it doesnt really matter, I dont have the assignment anymore.

 

[imported_Seer]

do you mind posting back with the answers? I'm interested and too lazy to look it up.

 

[CycloWizard]

Originally posted by: ChineseGuy
E is the greek letter epsilon, my professor never really did defined it, he just gave us examples of it.. lets see if I can get the symbol to come up right...

k! ek (the second k is suppose to be a subscript) .... soo k factorial multiplied by the "epsilon sub 1,2..." soo it would be like k!*[1,0,0,0]T k!*[0,1,0,0]T ... where the 1 moves down to the next row... I dont know if I am making a whole lot of sense. Oh well, I just turned the assignment in, so I dont have it in front of me anymore. At this point it doesnt really matter, I dont have the assignment anymore.

Ah, so epsilon is something like the kth unit vector. In that case, I would think your solution approach is fine. You could simplify it a bit by integrating four times right up front, then solving the four constraint equations, but the result would be the same.

 

[ChineseGuy]

Hmm.. the professor emailed me back, I guess my approach is not right, but i got the idea of it. Since school is out, he would not be returning the assignment. Oh well, I guess this is one of those that we may never find a direct answer to. Thanks for all the help guys.