First order system DE -> second order

[tourjete]

first order system DE --> second order

Homework Statement

Find a second-order DE for x alone that is equivalent to this system.

Homework Equations

dx/dt = 2x-y

dy/dt = -x

The Attempt at a Solution

I honestly have no clue where to start; in class we pretty much only stuck to springs when discussing second order equations.

Do I have to integrate the two given equations with respect to t so I have the t's in the equation? Or should I differentiate so I have a second derivative and hence a second order equation?

 

[Dick]

It's not that complicated. For example, just solve the second equation for x and substitute that into the first equation.

 

[fzero]

You can differentiate the first equation and then use the 2nd to eliminate y.

 

[tourjete]

Thanks guys! I did what fzero sugested and just want to make sure I did it right. I solved the second equation for y so I could eliminate it from the other one and got y = -xt.

I then plugged that into the first equation to get dx/dt = 2x + xt + C. Differentiating with respect to t i gotthat the second derivative is x + C. Is this right? It seems simplistic given the derivatves that were given.

 

[fzero]

What I meant was take the derivative of the first equation to get

$$\frac{d^2x}{dt^2} = 2\frac{dx}{dt}-\frac{dy}{dt},$$

then use the 2nd to write this as

$$\frac{d^2x}{dt^2} = 2\frac{dx}{dt}+x.$$