Characteristic Equation for Asin(Wt) + Bcos(Wt): Explained

[Ry122]

http://users.on.net/~rohanlal/qM.jpg
I don't understand how this answer is obtained for the homogenous solution.
What does characteristic equation in "r" mean and how does it help achieve the final solution of Asin(Wt) + Bcos(Wt)?

 

[Cyosis]

They use a "guess" solution. They guess that the solution is of the form $e^{rt}$. Plug this "guess" solution into your homogeneous differential equation and you will see how to obtain the characteristic equation.

 

[Ry122]

what does x with two dots mean? why would u guess e^rt, isn't the form of the guess supposed to be the same as the form of the equation?

 

[Cyosis]

Dots are derivatives, two dots is a double derivative. Just plug in the "guess" solution I provided you and show me your work and results.

Define "form of the equation".

 

[nealh149]

Perhaps what the OP means is when you are solving for the nonhomogeneous part, that part of the solution takes the form of the forcing function.

 

[zcd]

Remember that e^(i*x) = cosx + i*sinx, so e^(i*wt) = ? and e^(i*-wt) = ?

 

[HallsofIvy]

This is very peculiar! You are apparently trying to do a differential equation but don't seem to know the basics even of calculus. "x with 2 dots over it" is a standard notation for the second derivative; x" and dx/dt are also often used. "Assuming" a solution of the form [itex]e^{rt}[/tex] (even though in this case, it isnt', strictly speaking) is a standard method of arriving at the "characteristic equation" of the differential equation.

You may be trying to read a physics book that is assuming you know more mathematics than you do. (The dot notation is more common in physics than mathematics.) If so, either skip over the "mathematics" parts or start with a more basic physics book.

(I just realized this is posted under "homework". Talk to your teacher about what leve of mathematics you are expected to know.)