System of 3 Linear DEs in three variables-elimination

In summary, the problem involves solving a system of 3 linear differential equations in three variables using systematic elimination. The equations are Dx = y, Dy = z, and Dz = x. To solve this, the first step is to write the equations as Dx - y = 0, Dy - z = 0, and Dz - x = 0. Then, by taking two equations and eliminating one variable, and repeating this process with the remaining equations, you can obtain a new equation with only one variable. In this case, the variable y is eliminated from the equations D^2y - Dz = 0 and -x + Dz = 0, resulting in D^3y - x =
  • #1
bcjochim07
374
0
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

 
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  • #2


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? :wink:
 
  • #3


bcjochim07 said:

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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  • #4


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.
 

FAQ: System of 3 Linear DEs in three variables-elimination

What is a system of 3 linear differential equations in three variables?

A system of 3 linear differential equations in three variables refers to a set of three equations that involve derivatives of three variables with respect to a common independent variable. These equations are linear, meaning the highest power of the variables in each equation is 1, and they are interconnected, meaning the variables appear in more than one equation.

How do you solve a system of 3 linear differential equations in three variables through elimination?

The elimination method involves manipulating the equations to eliminate one of the variables. This is done by adding or subtracting the equations in a way that cancels out one of the variables. Once one variable is eliminated, the system can be solved by using substitution or other methods.

What is the purpose of solving a system of 3 linear differential equations in three variables?

Solving a system of 3 linear differential equations in three variables allows us to determine the values of the variables at a given point in time. This is useful in many scientific fields, such as physics, engineering, and economics, where equations involving rates of change are commonly used to model real-world phenomena.

Are there any limitations to the elimination method for solving a system of 3 linear differential equations in three variables?

Yes, the elimination method may not work for all systems of equations. In some cases, it may not be possible to eliminate a variable or the resulting equations may be difficult to solve. In these situations, other methods such as substitution or matrix operations may be more effective.

Can the elimination method be used for systems of more than 3 linear differential equations?

Yes, the elimination method can be extended to systems with more than 3 equations and variables. However, as the number of equations and variables increase, the process becomes more complex and time-consuming. In these cases, other methods such as matrix operations or numerical methods may be more efficient.

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