Undetermined Coefficients / Variation of Parameters

[amcavoy]

I know how to solve the following ODE with variation of parameters:

$$y''+4y=4\sec{\left(2t\right)}.$$

Is there any way to solve this with undetermined coefficients? So far I have tried Y_p=Acos(2t)+Bsin(2t), but that didn't work.

Thanks for the help.

 

[amcavoy]

Now that I try different methods, it seems that reduction of order won't work. At least, I come up with a function I can't integrate:

Y=vy_h

v'=u

$$u=\frac{2c_1t-2c_2\ln{\left(\cos{2t}\right)}}{c_1\cos{2t}+c_2\sin{2t}}$$

 

[saltydog]

I know how to solve the following ODE with variation of parameters:
$$y''+4y=4\sec{\left(2t\right)}.$$
Is there any way to solve this with undetermined coefficients? So far I have tried Y_p=Acos(2t)+Bsin(2t), but that didn't work.
Thanks for the help.

The method of undetermined coefficients is applicable only if the RHS of the non-homogeneous equation is itself a particular solution of some homogeneous linear differential equation with constant coefficients. Since Sec(2t) is not such a solution, this method is not applicable.