Question

I apologize if my question is not super specific. I just watched lecture 13 which is on finding particular solutions to inhomogeneous equations. In the video he mentions functions of importance, which were e^ax, sin(wx), cos(wx), e^(ax)sin(wx), e^(ax)(cos(wx)). In all of these cases, as far as I understand, he rewrote them in the form e^(a+iw). I did not in the lecture see any cases where you have something like t*e^t+4. I am not sure how I would get it in that e to the whatever form or even if it is possible. Is the technique even valid in that case?

Answer 1

I realized I'm not qualified to make a proper explanation of a technique.. :d

Answer 2

A general thought is to apply the laws of logarithms though.

Answer 3

e^(ln(t))*e^(t)+4

Answer 4

This means that t>0 though; so it might falsify the solutions in the end. If you're supposed to use imaginary stuff I got no clue as to how they apply in combination with exponential functions and their inverses.

Answer 5

Okay, so does the constant part mess up the e^(at) or is it okay

Answer 6

I'm not sure I follow what you mean by mess up the e^(at) :D

Answer 7

sorry. In the lecture he said that you can rewrite your f(x) as e^((a+iw)x). When I mentioned a I meant the complex factor and the 't' was the x. So I was wondering if the f(x) portion was okay being something like e^(a+iw)x + g, where g is a constant.

Answer 8

I stand by what I said in the first comment; I'm not qualified enough to say something. Hehe.

Answer 9

Thanks. I wouldn't have thought of the log rules thing.