- #1
skrat
- 748
- 8
Homework Statement
Solve: ##\ddot{x}+\Omega^{2} x=D+\frac{C}{2}+Ecos\omega t+\frac{C}{2}cos2\omega t##
Homework Equations
The Attempt at a Solution
I got a hint to use ##x=\alpha sin\omega t+\beta cos\omega t## so ##\ddot{x}=-\alpha ^{2}\omega ^{2}sin\omega t-\beta ^{2}\omega ^{2}cos\omega t## in the equation above than:
##(-\alpha ^{2}\omega ^{2}sin\omega t-\beta ^{2}\omega ^{2}cos\omega t)+\Omega^{2}x=\alpha sin\omega t+\beta cos\omega t=D+\frac{C}{2}+Ecos\omega t+\frac{C}{2}cos2\omega t##
Which gives me 4 separate equations depending on ##sin\omega t##, ##cos\omega t##, ##cos2\omega t## and constant:
first: ##-\alpha ^{2}\omega ^{2}+\Omega ^{2}\alpha=0##
second: ##-\beta ^{2}\omega ^{2}+\Omega ^{2}\beta =E##
third: ##\frac{C}{2}=0##
fourth: ##D+\frac{C}{2}=0##
Forth and third together say that ##D=0## and ##C=0##
First says that:
##\alpha ^{2}\omega ^{2}=\Omega ^{2}\alpha##
##\alpha =(\frac{\Omega }{\omega })^{2}##
But for second I am not sure, whether I can divide it with ##\beta## (probably not since it could be equal to 0) or how do I solve it?
PLEASE HELP
BTW, if everything is completely wrong and this is not how usually this kind of equations are solved, please let me know.