Rewriting a Second Order Equation as a System of First Order
Equations
To rewrite a second order equation as a system of first order equations, begin with,
m
d 2 y (t )
dy (t )
+c
+ ky (t ) = 0
2
dt
dt
m&y&(t ) + cy& (t ) + ky(t ) = 0
or
and initial conditions
y (t 0 ) = y0 , y& (t 0 ) = v0
Where x(t) is the vertical displacement of the mass about the equilibrium postion. m is
the mass, c is the damping constant, and k is the spring constant. With m ≠ 0 , we can set
C = mc and K = mk and rewrite the model as:
&y&(t ) + Cy& (t ) + Ky (t ) = 0
Now, we let z1 (t ) = y (t ) and z 2 (t ) = y& (t ) . Now we can express,
z&1 (t ) = z 2 (t )
and
z& 2 (t ) = −Cz 2 (t ) − Kz (t )
Putting this together we can write,
⎡ z&1 ⎤
⎡ 0
⎢ z& ⎥ (t ) = ⎢− K
⎣
⎣ 2⎦
If we let
1 ⎤ ⎡ z1 ⎤
(t )
− C ⎥⎦ ⎢⎣ z 2 ⎥⎦
⎡z ⎤
z (t ) = ⎢ 1 ⎥ (t)
⎣ z2 ⎦
and
⎡ 0
A=⎢
⎣− K
1 ⎤
− C ⎥⎦
we can write the system as,
z&(t ) = Az(t ) with
⎡y ⎤
z ( 0) = ⎢ 0 ⎥
⎣ v0 ⎦