# Method of Undermined coefficients

I am working on the differential equation

(dx^2)/(dt^2) + dx/dt + x =sin (Ï‰t)

I have found the general solution of m^2+m+1=0 which is

x=e^(-1/2)t(Acos((sqrt3)/2)t+Bsin((sqrt3)/2)t

I am looking for a particular integral that satisfies the differential equation so as to obtain the general solution

I am finding great difficulty in finding the particular integral xp(t) through the method of undetermined coefficients because i seem to not be able to guess the form of the solution properly probably to a failure of basic maths ability

I would appreciate a few words on the thought process to arrive at how we make a guess of the form of the solution

so far (and i am not sure this is right) I have

k=Csin(Ï‰t)

and differentiate twice

k'=CÏ‰cos(Ï‰t)

k''=-CÏ‰^2sin(Ï‰t)

I then substitute into left hand side

we get

-CÏ‰^2sin(Ï‰t)+CÏ‰cos(Ï‰t)+Csin(Ï‰t)=sin(Ï‰t)

which gives

-CÏ‰^2+C=1

and

CÏ‰=0

so by now i am completely lost

please help me to understand how to arrive at the general solution

Thanks

https://brainmass.com/math/basic-algebra/method-undermined-coefficients-562968

## SOLUTION This solution is **FREE** courtesy of BrainMass!

Hello and thank you for posting your question to Brainmass.

The solution is attached below in two files. the files are identical in content, only differ in format. The first is in MS Word format, while the other is in Adobe pdf format. Therefore you can choose the format that is most suitable to you.

In the first three pages I simply re-solve the homogeneous equation and get the same homogeneous solution as yours.

In pages 4-6 I solve for the particular solution in the method of undetermined coefficients, including a MAPLE verification of the solution.

In pages 7-10 I show how to use the method of variation of parameters. I show how to get the integrals (but I do not solve them) for the particular solution. The final integral are doable, but they are just tedious (integration by parts twice of exponents and trigonometric functions is not fun). However, you may want to keep it for future reference.

https://brainmass.com/math/basic-algebra/method-undermined-coefficients-562968