Reduction of Order Non-homogenous System

Question

I have to write the following system as a first order system of differential equations

x'''-2x''+4x'-y= sin(t)

y''-2y+x''= cos(t)

I have no idea where to start.

Answer 1

$$x'''-2x''+4x'-y= \sin(t)\quad (1)$$

$$y''-2y+x''= \cos(t)\quad (2).$$

Let $$x_1=x\quad (3)$$ $$x_2=x'\quad (4)$$ $$x_3=x''\quad (5)$$ $$y_1=y\quad (6)$$ $$y_2=y'\quad (7).$$ Then the above system is reduced to a first order system $X'=AX+B(t)$ as follows. $$(3), (4)\Rightarrow x_1'=x_2$$ $$(4), (5)\Rightarrow x_2'=x_3$$ $$(6), (7)\Rightarrow y_1'=y_2$$ $$(1)\Rightarrow x_3'=2x_3-4x_2+x_4+y_1+\sin t$$ $$(2)\Rightarrow y_2'=-x_3+2y_1+\cos t,$$ where $X=(x_1, x_2, x_3, y_1, y_2)^t$ and $B(t)=(0, 0, \sin t, 0, \cos t)$.

Answer 2

Write the system in $u_0=x$, $u_1=x'$, $u_2=x''$ and $u_4=y$, $u_5=y'$ with the additional relations like $u_1'=u_2$,...