Rewriting second order DE as first-order system in "normal form"

Question

I have the following problem: "Rewrite the given scalar equation as a​ first-order system in normal form. Express the system in the matrix form $x'=Ax+f$, Let $x_1(t)=y(t)$ and $x_2(t)=y'(t)$

$y''(t)-8y'(t)-5y(t)=tan(t)$"

However, I'm not sure what a "system in normal form" is. I don't see it in the online textbook when I ctrl+f it...

Answer 1

Hint

you have $x_1=y$ then $\dot x_1=y'=x_2 $ $$\implies \dot x_1=\color{red}{0x_1}+\color{blue}{1x_2}$$

You also have $y''= \dot x_2 =8y'+5y+ \tan(t)=8x_2+5x_1 +\tan(t)$ $$ \implies \dot x_2=\color{blue}{5x_1}+\color{blue}{8x_2}+\tan(t)$$

Therefore

$$\pmatrix { \dot x_1 \\ \dot x_2}=\pmatrix {0 & 1 \\ 5 & 8}\pmatrix{ x_1 \\x_2}+ \pmatrix {0 \\ \tan(t)}$$ $$\dot x=Ax+f$$