Homogenous Linear Systems with Constant Coefficients:  Find a general solution to the system of equations and describe the behavior as `t -> oo` Then, draw a direction field and plot just a...

Homogenous Linear Systems with Constant Coefficients: 

Find a general solution to the system of equations and describe the behavior as `t -> oo`

Then, draw a direction field and plot just a few trajectories of the system.

`x' = ((1, -2), (3, -4))x`

1 Answer | Add Yours

sciencesolve's profile pic

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

I think that you should provide the following information for the system, such that:

`[[x'],[y']]` = `[[1,-2],[3,-4]]` *`[[x],[y]]`

Hence,

`x' = x - 2y`

`y' = 3x - 4y`

You may eliminate x, by solving the first equation for x, such that:

`x = x' + 2y`

Replace `x' + 2y` for x, in the second equation, such that:

`y' = 3x' + 6y - 4y`

`y' = 3x' + 2y`

`y'' - 3y' + 2y = 0`

Replacing r for y, yields:

`r^2 - 3r+ 2 = 0`

`r_(1,2) =(3+-sqrt(9 - 8) )/2`

`r_(1,2) = (3+- 1)/2`

`r_1 = 2`

`r_2 = 1`

Hence, `y = c_1*e^(r_1t) + c_2*e^(r_2t)`

`y = c_1*e^(2t) + c_2*e^t`

`y' = 2c_1*e^(2t) + c_2*e^t`

`y' - 2y = 3x' => 2c_1*e^(2t) + c_2*e^t - c_1*e^(2t) + c_2*e^t = 3x'`

`3x' = c_1*e^(2t) + 2c_2*e^t`

`x = x' + 2y => x = (c_1*e^(2t) + 2c_2*e^t)/3 + 2c_1*e^(2t) + 2c_2*e^t`

`x = (c_1*e^(2t) + 2c_2*e^t + 6c_1*e^(2t) + 6c_2*e^t)/3`

`x = (7c_1*e^(2t) + 8c_2*e^t)/3`

Hence, the solution to the  given system is, such that `x = (7c_1*e^(2t) + 8c_2*e^t)/3 and y = c_1*e^(2t) + c_2*e^t.`

We’ve answered 318,917 questions. We can answer yours, too.

Ask a question