Solving ODE's or Euler second order diff. eq's containing Asecx?

[hivesaeed4]

I know how we solve ODE's and euler equations in which we have cos and/or sin terms on the right. We take the particular solution to be Acos(x) + Bsin(x). But what if we have secant or cosecant terms on the right or tan and/or cotangent terms?

Qno. 1 Are these 4 terms possible i.e. can they come in non-homogenous ODE's or euler eq's ?

Qno. 2) If yes, then how do we solve them?

 

[Mute]

Typically, when you are first introduced to non-homogeneous differential equations, the forcing function is something simple like a constant, t^n, exp(-t), cos(t), sin(t), etc. For such forcing functions it's typically easy enough to guess what sort of form the particular solution should have, and then it's a matter of finding out the coefficients. This is the "Method of Undetermined Coefficients" (or, as a prof of mine once put it, "The Method of Educated Guessing").

If you have more complicated forcing functions like sec(x) or tan(x), it's not as obvious what you should guess, so you have to use a more general method.

Two such methods are Variation of parameters or Green's function approaches.