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

Let so

Substitute. This gives

Re-grouping gives

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

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