Solving a Two-Level System of Differential Equations with Laplace Transform

[indigojoker]

Homework Statement

How would I solve a two-level system such as this:

y_1'=-ay_1+by_2

y_2'=-ay_2-by_1+c

where a b and c are constants.

Homework Equations

Laplace transform

The Attempt at a Solution

I guess my question is what method would I use to solve this? As in should I solve for y1 in terms of y2 and then plug that back into the second equation? Not too sure about the approach.

 

[Glass]

I would do exactly as you suggested - solve for y2 in terms of y1 and sub back in. You will get a 2nd order ODE, but it is not too difficult. Moreover, if your unsure about the answer, you can always plug your solution back into the equation to ensure your solution satisfies the DE (although this doesn't show you have found all possible solutions). That's one of the great things about them - it's pretty easy to check if your solution is correct.

 

[HallsofIvy]

Homework Statement

How would I solve a two-level system such as this:

y_1'=-ay_1+by_2

y_2'=-ay_2-by_1+c

where a b and c are constants.

Homework Equations

Laplace transform

The Attempt at a Solution

I guess my question is what method would I use to solve this? As in should I solve for y1 in terms of y2 and then plug that back into the second equation? Not too sure about the approach.

I don't see any way to "solve for y1 in terms of y2" from the first equation because of the y_1' term, nor do I see any reason to use the Laplace transform for such a simple problem. There are, however, several methods you could use. One is to convert from two first order equations to one second order equation: Differentiate the first equation to get y_1"= -ay_1'+ by_2'. Now replace that y_2' with it's expression in the second equation: y_1"= -ay_1'+ b(-cy_1+ dy_2)= -ay_1'- bcy_1+ d(by_2). From the first equation again, by_2= y_1'+ ay_1 so the equation becomes y_1"= -ay_1'-bcy_1+ dy_1'+ ady_1 or y_1"+(a-d)y_1'+ (ad- bc)y_1= 0[/itex]. Solve that equation for y_1 and then use by_2= y_1'+ ay_1 to solve for y_2.

Another way is to write it as a matrix equation:
If you let
Y= \left[\begin{array}{c}y_1 \ y2 \end{array}\right]
then your equation becomes
Y'= \left[\begin{array}{cc}a & b \ c & d\end{array}\right]Y
You can solve that by finding the eigenvalues and corresponding eigenvectors of the matrix.