Thread: 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!

Consider



Let so




Substitute. This gives



Re-grouping gives



Now pick such that . This gives an ODE absent of the term .

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

Only for certain because what you want is



to become



meaning that both and must be satisfied. However, why go to all that trouble. If you have



then letting gives a linear ODE in