Yahoo Answers is shutting down on May 4th, 2021 (Eastern Time) and beginning April 20th, 2021 (Eastern Time) the Yahoo Answers website will be in read-only mode. There will be no changes to other Yahoo properties or services, or your Yahoo account. You can find more information about the Yahoo Answers shutdown and how to download your data on this help page.
Trending News
Differential equations problem, find the general solution of this equation: y'' + 4y = sec(2t)?
I just need to know the method to use! and maybe the first few steps to it, i dont need it completely worked out... unless you want to :]]]. but yeah my first thought was to use the method of undetermined coefficients, but i cant see how the "sec(2t)" would work with that? any help?? thanks :]
1 Answer
- MechEng2030Lv 71 decade agoFavorite Answer
No, undetermined coefficients will definitely not work here. Use variation of parameters.
Solve for the homogeneous system first:
y'' + 4y = 0
y=e^(rt)
r² + 4 = 0
r = +/- 2i
y_h = C_1*sin(2t) + C_2*cos(2t)
y_1=sin(2t),y_2=cos(2t)
u_1'y_1 + u_2'y_2 = 0
u_1'y_1' + u_2'y_2' = sec(2t)
u_1'*sin(2t) + u_2'*cos(2t) = 0
2u_1'*cos(2t) - 2u_2'*sin(2t) = sec(2t)
Solve for either u_1' or u_2'. I'll solve for u_1'.
u_1' = 1/2
u_1 = t/2 and u_2 = 1/4*ln|cos(2t)|
GENERAL SOLUTION:
y = C_1*sin(2t) + C_2*cos(2t) + t/2*sin(2t) + 1/4*ln|cos(2t)|*cos(2t)
