Question

Transform the given equation into a system of first order equations. (Let x1 = u, x2 = u', x3 = u'', and x4 = u'''. Enter your answers in terms of x1, x2, x3, and x4.) u(4) − u = 0

a) x1' =
b) x2' =
c) x3' =
d) x4' =

Answer 1

To transform the given equation into a system of first-order equations, we can use the definitions x1 = u, x2 = u', x3 = u'', and x4 = u'''. The corresponding system of equations is: a) x1' = x2, b) x2' = x3, c) x3' = x4, d) x4' = 0.

Step-by-step explanation:

To transform the given equation into a system of first-order equations, let's define the variables as follows: x1 = u, x2 = u', x3 = u'', and x4 = u'''. The given equation is u(4) - u = 0. To express this equation in terms of the new variables, we have:

a) x1' = x2

b) x2' = x3

c) x3' = x4

d) x4' = 0

Answer 2

Given the 4th order linear ODE

`u^((4)) - u = 0`

we substitute

`x_1 = u`

`x_2 = {x_1}' = u'`

`x_3 = {x_2}' = {x_1}'' = u''`

`x_4 = {x_3}' = {x_2}'' = {x_1}''' = u'''`

Then the given equation transforms to

`{x_4}' - x_1 = 0`

but we also need to relate this to the other derivative substitutions. This gives the system of differential equations

`\begin{cases} {x_1}' = x_2 \\ {x_2}' = x_3 \\ {x_3}' = x_4 \\ {x_4}' = x_1 \end{cases}`

In matrix form,

`\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix}' = \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix}`