transform out 1st order term from 2nd order linear ode
Hello,
I am a PHD student in engineering with a strong background in ODEs and PDEs.
I am working through a text called "Combustion Physics" by C. K. Law. On p.327, Law says:
"It is, however, well known that in a second order linear ordinary differential equation, the first order differential can be transformed away."
Such a technique is NOT well known to me and I cannot find anything like it in my undergrad ODEs text (Boyce and DiPrima). Unfortunately, Law doesn't provide a reference, nor does the example he provides explain the method. Can someone refer me to a book where I can learn more about this?
Thanks!
Re: transform out 1st order term from 2nd order linear ode
Consider
 y' + q(x) y = f(x))
Let  u)
so
Substitute. This gives
 + q a u = f)
Re-grouping gives
u' + \left(a'' + pa' + qa \right)u = f)
Now pick 
such that 
. This gives an ODE absent of the term 
.
Re: transform out 1st order term from 2nd order linear ode
Danny,
Ah, a clever solution! What if, following your nomenclature, q(x) = 0 and in the end, I wish to have only a second order term? In other words, is there a method by which I can eliminate the first order differential from
y" + p(x) y' = f(x)
leaving only a second order differential and no u term?
Thanks,
T
Re: transform out 1st order term from 2nd order linear ode
Only for certain 
because what you want is
u' + \left(a'' + pa' \right)u = f)
to become
meaning that both and 
must be satisfied. However, why go to all that trouble. If you have
 y' = f(x))
then letting 
gives a linear ODE in 
