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 e^{rt}[/tex] (even though in this case, it isnt&#039;, strictly speaking) is a standard method of arriving at the &quot;characteristic equation&quot; of the differential equation.<br /> <br /> 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 &quot;mathematics&quot; parts or start with a more basic physics book.<br /> <br /> (I just realized this is posted under &quot;homework&quot;. Talk to your teacher about what leve of mathematics you are expected to know.)