# Eigenvectors/values and diff equations

• EvLer
In summary, the higher order equation y'' + y = 0 can be written as a first order system by introducing y' as another unknown besides y. This can be represented by the matrix equation d[y y']t/dt = [y' y'']t = [y' -y]t, which can be simplified to du/dt = Au. The 2x2 matrix A for this equation is [0 1; -1 0]. This may result in complex eigenvalues, but this does not affect the validity of the solution. The problem also asks to find the eigenvectors and compute the solution starting from y(0) and y'(0)=0, which can be done using the formulas for

## Homework Statement

the higher order equation y'' + y = 0 can be written as a first order system by introducing y' as another unknown besides y:

## Homework Equations

d[y y']t/dt = [y' y'']t = [y' -y]t

if this is du/dt = Au what is 2x2 matrix A?

## The Attempt at a Solution

I think it's
[0 1]
[-1 0]
but then I get complex eigenvalues so I am a bit suspicious... could someone verify or help me out?
thanks.

You are exactly correct. If you let x= dy/dt, then d2y/dt2= dx/dt+ y= 0 or dx/dt= -y so your two equations are dy/dt= x and dx/dt= -1. Letting Y(t)= (y(t) x(t))t, yes, your differential equation is
$$Y(t)= \left[\begin{array}{cc} 0 & 1 \\ -1 & 0\end{array}\right]Y(t)$$
and, yes, that has complex eigenvalues. Why does that bother you?
I notice that the problem only asks you to write the matrix equation, not to find the eigenvalues or solve the equation so I guess you will learn what complex eigenvalues mean in this situation later.

I'm trying to finish the very same problem. In addition to what is stated above, the problem says: "Find its eigenvectors, and compute the solution that starts from y(0), y'(0)=0."

I found the eigenvalues to be i and -i, and the eigenvectors to be
[1 -i]^T and [1 i]^T.

I wrote the solution to start as y(t)=(1/2)*e^it*[1 i]^T + (1/2)*e^-it*[1 -i]^T.

From there I don't know how to proceed. Any suggestions?

eit = cos(t) + isin(t)

Note that (1/2)(eit + e-it) = cos(t)
and that (1/(2i))(eit - e-it) = sin(t)