System of 3 Linear DEs in three variables-elimination

[bcjochim07]

System of 3 Linear DEs in three variables--elimination

Homework Statement

Solve the given system of linear DEs by systematic elimination.

Dx = y
Dy = z
Dz = x

What I wanted to do is solve this like you would any other system of three eqns, so I wrote:

Dx - y + 0z = 0
0x +Dy - z = 0
-x +0y +Dz = 0

and then I attempted to take two of the equations and eliminate one variable and take another two and eliminate the same variable and then combine those two. But this doesn't work because in each equation only two of the variables are present. Any pointers would be greatly appreciated. Thanks.

Homework Equations

The Attempt at a Solution

 

[gabbagabbahey]

The key phrase here is "systematic elimination"...let's look at your first two equations:

(1)Dx=y and (2)Dy=z...how could you go about eliminating 'y' from both of these equations? hint: what is D^2x?

 

[HallsofIvy]

Homework Statement

Solve the given system of linear DEs by systematic elimination.

Dx = y
Dy = z
Dz = x

What I wanted to do is solve this like you would any other system of three eqns, so I wrote:

Dx - y + 0z = 0
0x +Dy - z = 0
-x +0y +Dz = 0

and then I attempted to take two of the equations and eliminate one variable and take another two and eliminate the same variable and then combine those two. But this doesn't work because in each equation only two of the variables are present.

In other words, one variable has already been eliminated from one equation- part of your work has already been done! Just eliminate that variable from the other two equations. For example, if you decided to eliminate z, notice that z does not appear in the first equation. So you only need to eliminate z from equations 2 and 3: Dy= z and Dz= -y. As gabbagabbahey suggested, Differentiate the equation Dy- z= 0 to get "Dz" and replace "Dz" in the last equation.
Any pointers would be greatly appreciated. Thanks.

Homework Equations

The Attempt at a Solution

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[bcjochim07]

Here's what I tried

eliminating z from the last two equations:

D^2y - Dz = 0
-x + Dz = 0

= D^2y - x = 0

Then I tried to add that to the first eqn.

-y + Dx
D^3y + -Dx

(D^3 - 1)y = 0
y = c1e^t + c2te^t + c3t^2e^t, but I know I must have done something wrong because y in my textbook has trig functions in it.
I don't think I am understanding what you are saying I should do.