Help Setting up an Equation to use the Elimination Method

In summary, the given problem is to solve the initial value problem (IVP) x'=2x+y-e^2t and y'=x+2y, where x(0)=1 and y(0)=-1. After rearranging the equations and substituting in the derivative operator D, the equations can be simplified to (D-2)x-Dy=-2e^2t and (D-2)y-Dx=0. The student's initial attempt was to eliminate the y variable, but they encountered difficulty in setting up the equations. However, with some help, they were able to solve the problem by multiplying one equation by D and the other by (D-2). This resulted in the simplified equation Dx
  • #1
daedie
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Homework Statement



The question is to solve the IVP: x'=2x+y-e^2t & y'=x+2y, where x(0)=1, y(0)=-1

Homework Equations



Arranging the equations and substituting in D for the derivatives, the equations become:
1. (D-2)x-Dy= -2e^2t
2. (D-2)y-Dx= 0

The Attempt at a Solution



My first attempt was to eliminate the y variable and leave x to solve for. But, looking at the problem, I'm having an issue with figuring out how to set the equation up in order to do so. One attempt was to eliminate the (D-2)x & y on both equations:

(D-2)x-Dy=-2e^2t *(D-2)y
(D-2)y-Dx=0 *(D-2)x

(D-2)x(D-2)y-Dy(D-2)y=(D-2)-2e^2t
(D-2)x(D-2)y-Dx(D-2)x=0

Subtracting, this leaves:

Dx(D-2)x-Dy(D-2)y=0

This is far messier than we've dealt with in class, but not beyond the realm something the teacher might give us. I'm wondering if there's an easier way to clear out one of the terms in order to make the Diff Eq easy to solve for x. Once I get one value solved, I can go back and figure out the other. Just the set up is tricky. Thanks for any help you can provide!

EDIT: I was able to get some help. Multiply (1) by D and (2) by (D-2) and voila!
 
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  • #2
Note that you wrote "y'=x+2y" as the second equation, and then proceeded to work with "2. (D-2)y-Dx= 0".
So somehow you have introduced a derivative operator on the x?
 

What is the Elimination Method?

The Elimination Method is a mathematical process used to solve systems of equations with two or more variables. It involves manipulating the equations to eliminate one variable, allowing for the remaining variable to be solved.

When should I use the Elimination Method?

The Elimination Method is best used when the coefficients of one of the variables in the system of equations are the same or have a common multiple. This makes it easier to eliminate that variable and solve for the other.

What are the steps to using the Elimination Method?

The steps to using the Elimination Method are as follows:
1. Identify the variable to eliminate
2. Multiply one or both equations by a number to make the coefficients of the variable the same or a multiple of each other
3. Add or subtract the equations to eliminate the variable
4. Solve for the remaining variable
5. Check your solution by plugging it back into the original equations

Can the Elimination Method be used on systems of more than two equations?

Yes, the Elimination Method can be used on systems of any number of equations. However, as the number of equations increases, the process can become more time consuming and complex.

Are there any limitations to using the Elimination Method?

Yes, the Elimination Method can only be used on systems of equations that have the same number of variables. It also only works if the equations are linear (no exponents or variables multiplied together).

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