Conversion Of A n-th order ODE Into A First-Order System of ODEs

From Problem Set 8: Exercise 1.3

Considering an nth-order ODE given as,

Here the parenthesis denotes the derivative and not the solution

By taking successive terms to be derivatives as

.... . We can plug there in to get an ODE in the first order ODE.

Thus

We can come to the system of

Thus we get

Thus we can get a system of successive derivatives and the system of ODEs. These are all in the first order from the nth order.

For example, if we have the equation of , we can set u = y and v = y'

to get the equations of and .