Solving y''+3iy'+y=cos(2t) with Undetermined Coefficients

[cragar]

If I wanted to solve this [itex] y''+3iy'+y=cos(2t) [/itex] using undetermined coefficients.
and I make the guess [itex] y=Ae^{2it} [/itex]
then i find y' and y'' and then solve for A. I get that A=-1/9
then I take the real part when I multiply it to Eulers formula.
But when I plug this back into check it doesn't work.
Is there something weird going on because I have an imaginary coefficient in front of the y'.

 

[JJacquelin]

If I wanted to solve this [itex] y''+3iy'+y=cos(2t) [/itex] using undetermined coefficients.
and I make the guess [itex] y=Ae^{2it} [/itex]
then i find y' and y'' and then solve for A. I get that A=-1/9
then I take the real part when I multiply it to Eulers formula.
But when I plug this back into check it doesn't work.
Is there something weird going on because I have an imaginary coefficient in front of the y'.

Better, try : y = A exp(2it) +B exp(-2it)
with cos(2t) = (1/2)exp(2it) + (1/2)exp(-2it)

 

[jackmell]

How about even more better:

[tex]y_p=A\cos(2t)+B\sin(2t)[/tex]

slap it in (the DE), equate coefficients, bingo-bango.

Also, nothing (algebraic) changes if not only the coefficients are complex but everything else like y and x are complex too. So don't let the i thing intimiate you. Just use regular ordinary complex arithemetic and muscle-through the algebra like always.

 

[cragar]

okay thanks for the advice

 

[blue_raver22]

It is possible that there is an error in your calculations or in the way you are checking the solution. It is important to carefully check each step of the process to ensure accuracy. Additionally, it is possible that the solution you have found is a particular solution, but not the general solution to the differential equation. It may be necessary to use other methods, such as variation of parameters, to find the general solution. It is also worth noting that the presence of an imaginary coefficient in front of y' does not necessarily indicate that something weird is going on. This is a common occurrence in differential equations and can be handled using standard techniques. It is important to carefully follow the steps of the method and double check your work to ensure an accurate solution.