Undetermined Coefficient Method: y''(x) + y(x) = 4xcos(x)
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Re: Undetermined Coefficient Method Help!
If you believe there to be only one root, you need to rethink.
\(\displaystyle r^{1} + 1 = 0 \implies r = i\) or \(\displaystyle r = -i\)
This should lead you directly to \(\displaystyle A\cos(x) + B\sin(x)\) as a good palce to start.
After that, it is time to imagine where that right hand side could come from. It appears that \(\displaystyle Cx\cos(x) + Dx\sin(x)\) might be a good try.
Sadly, this leads us nowhere and we must get a little more complicated. Maybe, \(\displaystyle Cx\cos(x) + Dx\sin(x) + Ex^{2}\cos(x) + Fx^{2}\sin(x)\)?
After two derivatives and some algebra, I get C = 1, D = 0, E = 0, and F = 1. No information is obtained for A or B. We'll need some initial values! Since we're not doing an Initial Value problem, we seem to be done.
It's a matter of guessing what the answer will look like and building it. The solution process can get a little heavy on the algebra. Go with courage!
If you believe there to be only one root, you need to rethink.
\(\displaystyle r^{1} + 1 = 0 \implies r = i\) or \(\displaystyle r = -i\)
This should lead you directly to \(\displaystyle A\cos(x) + B\sin(x)\) as a good palce to start.
After that, it is time to imagine where that right hand side could come from. It appears that \(\displaystyle Cx\cos(x) + Dx\sin(x)\) might be a good try.
Sadly, this leads us nowhere and we must get a little more complicated. Maybe, \(\displaystyle Cx\cos(x) + Dx\sin(x) + Ex^{2}\cos(x) + Fx^{2}\sin(x)\)?
After two derivatives and some algebra, I get C = 1, D = 0, E = 0, and F = 1. No information is obtained for A or B. We'll need some initial values! Since we're not doing an Initial Value problem, we seem to be done.
It's a matter of guessing what the answer will look like and building it. The solution process can get a little heavy on the algebra. Go with courage!
Re: Undetermined Coefficient Method Help!
(At2+Bt+C)cost+(Dtsquare+Ft+G)sint=yp an the roots that you found are included in the general solution.the complimentry solution is Kcost+Lsinx=yc. the sum of yc and yp is the general solution.the first derivative of the yp is taken and then second one is derived. yp and yp'' are summed.at last A=-F D=1=B A=0=F are acquired.the yp equation =tcost +tsquare sint . good luck
(At2+Bt+C)cost+(Dtsquare+Ft+G)sint=yp an the roots that you found are included in the general solution.the complimentry solution is Kcost+Lsinx=yc. the sum of yc and yp is the general solution.the first derivative of the yp is taken and then second one is derived. yp and yp'' are summed.at last A=-F D=1=B A=0=F are acquired.the yp equation =tcost +tsquare sint . good luck