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