# Re-write as a system of first order ODEs

1. May 30, 2012

### anonymity

hello,

I am going through the first chapter (a review chapter) of a second-course book in ODEs, and can't seem to remember how to re-write higher order DEs into a system of first order linear ODEs, and my old textbook only shows this for second order equations...

The question is: "Write the following differential equations as a system of first order ODEs:

y'' -5y'+6y=0
-y''-2y' = 7cos(y')
y(4) - y'' + 8y' + y2 = ex "

2. May 30, 2012

### algebrat

How about letting $y=y_0, y'=y_1$ et cetera?

3. May 30, 2012

### GaussGreyjoy

You have to make a change of variable, you can call y'=z and by doing the substitution you obtain a first order system of ODE.

Sorry for my English and I'll do it later from my computer if I haven't explained myself properly.

4. May 30, 2012

### SammyS

Staff Emeritus
Add the first two equations to eliminate y'' .

Is the really cos(y') and not cos(y) ?

Getting rid of the 4th derivative may be more difficult. I think you can differentiate the first two, one of them once, the other twice to eliminate y(4) (and the y''' you'll introduce).

Alternatively: If that's a cos(y), then you could combin the first two equations to eliminate y'. Differentiate that twice to get an additional equation with y(4) and y' and y. Combine that with the third equation to eliminate y(4) .