Re-write as a system of first order ODEs

[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 = e^x "

 

[algebrat]

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

 

[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.

 

[SammyS]

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 = e^x "

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⁾ .