# Meaning of “Canonical System of the First Order”

I am learning about PDEs and came across the following.

"Convert a partial differential equation of higher order into a canonical system of the first order"

What does the above statement mean/imply? How does one convert, say, a second order PDE into a canonical system of the first order? A simple example (or online readings) to illustrate the process will be appreciated.

Regards, Radz.

• Do you know how to convert a higher order ODE into a first-order ODE? – Christopher A. Wong Jul 8 '14 at 9:51
• Dear @Chris A. Wong, Yes. – Radz Jul 8 '14 at 10:06
• The same principle applies. You create new variables that stand in for the higher derivatives of the function, and then you add new equations to the system that reinforce the condition that your new variable are the derivatives of the original function. – Christopher A. Wong Jul 8 '14 at 10:27
• Dear @Chris A. Wong, I think I understand what you mean. Could you please verify the following. Suppose that $u=u(x,y)$ and consider the 2nd order PDE $F(x,y,u,u_x,u_y,u_{xy},u_{yx},u_{xx},u_{yy})=0$. To convert to a system of 1st order PDEs, let $p=u_x$ such that $p_x=u_{xx}$ and $p_y=u_{xy}$. Similarly, let $q=u_y$ such that $q_y=u_{yy}$ and $q_x=u_{yx}$. The canonical system is $$p=u_{x},\quad q=u_{y},\quad P(x,y,u,p,q,p_y,q_x,p_x,q_y)=0.$$ Is this correct? – Radz Jul 8 '14 at 11:16
• I believe that looks correct. I don't know what is meant by "canonical", but it is certainly a first-order system. – Christopher A. Wong Jul 9 '14 at 6:53