Question

Given y'''+ay''+by'+cy=0

Transform the equation into a system of first order equations setting x1=y, x2=y', x3=y''.

Then write the system in vector matrix form x'=Ax

Answer 1

@wio

Answer 2

Basically do something like...

Answer 3

\[
x_1 = y\\
x_2 = y' \\
x_3 = y'' \\
x_4 = y''' \\
\]

Answer 4

i thought though that since it is a third order equation that there will only be 3 vectors

Answer 5

\[
x_1' = x_2\\
x_2' = x_3\\
x_3' = x_4\\
x_4 + ax_3 + bx_2 +cx_1 = 0\\
\]

Answer 6

so how do I get a 3x3 matrix from that?

Answer 7

Umm, substitute the last two equations.

Answer 8

\[
x_1' = x_2\\
x_2' = x_3\\
x_3' = x_4\\
x_4 = - ax_3 - bx_2 -cx_1\\
\]
So do this.
\[
x_1' = x_2\\
x_2' = x_3\\
x_3' = - ax_3 - bx_2 -cx_1\\
\]

Answer 9

\[\left[\begin{matrix}0 & 1 & 0 \\ 1 & 0 &0 \\-a & -b & -c\end{matrix}\right]\]

Answer 10

I guess anytime you have a third order homogeneous equation you can just plug that into the matrix and get the solutions without doing much of any work... is that what this is trying to tell me

Answer 11

Basically.

Answer 12

You can do it for any homogeneous linear equation

Answer 13

Even if it isn't a homogeneous equation, you need to solve the homogeneous one anyway. so it's helpful even when it's not homogeneous.

Answer 14

thats nice... definitely makes things easier on some of these problems. thank you again for your help.