# System of first order equations, matrix form, quick question

Question:

##h_{t}+vh_{x}+v_{x}h=0##
##v_{t}+gh_{x}+vv_{x}=0##

Write it in the form ##P_{t}+Q_{x}=0##, where ##P=(h,hv)^{T}##,
where ##g## is a constant ##>0##, and ##v## and ##h## are functions of ##x## and ##t##.

Attempt:

I have ##Q=(vh,?)^{T}##, the first equation looks easy enough,

but I'm unsure on the second equation as the component of ##P## for this is giving ##hv_{t}+h_{t}v##, whereas I need to get to ##v_{t}+gh_{x}+vv_{x}##,

So I'm thinking maybe we need to take a time/x derivaitve of the 2nd equation or multiply it by something. So far multiplication by ##h## seems most promising to me, but I still can't get it, doing this i need the ##Q_{x}## to yield: ##h_{t}v+gh_{x}h+vhv_{x}##, I'm pretty sure it's not possible to get the ##h_{t}v## term?

I am correct in thinking you can multiply one of the equations/ take derivatives of one of the equations and not the other right?

Thanks very much !

Multiply the first equation by v and the second by h and add both equations. Note that $(vh)_t = v_th + vh_t$. Some more hints: how can you rewrite $uu_x =(?)_x$? What is $(uvw)_x$ if you write it out?