Fundamental matrix vs Wronskian

[summerha]

I have just learned the first order system of ODE,

i found that the Wronskian in second order ODE is |y1 y2 ; y1' y2'|

but in first order system of ODE is the Wronskian is W(two solution),

i wonder which ones is the general form?

thank you very much

 

[HallsofIvy]

I'm not clear what you mean by "second order ODE" and "first order system".
I would interpret "first order system" to mean two first order differential equations such as y1'= ay1+ by2, y2'= cy1+ dy2. The "Wronkskian" of that is $\left|\begin{array}9 y1 & y2 \\ y1' & y2'\end{array}\right|= y1y2'- y1'y2$.

By 'second order ODE" I would understand a single second order equation: y''+ by'+ cy= 0. That can be converted to a system of first order equations by defining y1= y, y2= y'. Then y1'= y2 and y''= y2'= -by2+ cy1. The solutions to that system are the same as the solutions to the original second order equation so the Wronskian for that system can be considered to be the Wronskian for the original equation though I think that would be an unusual use of the word!

 

[summerha]

yes , this is what i want to ask, i am sorry for my unclear expression.

about the Wronkskian of two first order differential equations such as y1'= ay1+ by2, y2'= cy1+ dy2

the Wronkskian is not det(solution1 solution2)? (two set of solution column vector)