- #1
Lahooty
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Homework Statement
Solve the following systems by either substitution or elimination:
dx/dt = y
dy/dt = -x + cos(2t)
Homework Equations
I know the solution is:
x(t) = c_1cos(t) + c_2sin(t) - 1/3cos(2t)
y(t) = -c_1sin(t) + c_2cos(t) + 2/3sin(2t)
The Attempt at a Solution
x' = [ 0 1; -1 0][x; y] + cos(2t)[0; 1]
Det(A-λI) = [-λ 1; -1 -λ] = λ^2+1 = λ_1 = i, λ_2 = -i
λ = i; A-λi = [-i 1; -1 -i]
(i)x + y = 0
x = 1, y = -i;
v = [1; -i] = [1; 0] + i[0; -1]
x(t) = c_1*cos(t) + c_2*sin(t);
y(t) = c_1*sin(t) - c_2*cos(t);
[0 1; -1 0]*a = [0; -1]
a = [1; 0]
[0 1; -1 0]*b = [1; 0]
b = [0; 1]
x(t) = c_1*cos(t) + c_2*sin(t) + cos(2t);
y(t) = c_1*sin(t) - c_2*cos(t) + 1;
I used the Undetermined Coefficients method:
http://tutorial.math.lamar.edu/Classes/DE/RealEigenvalues.aspx#Ex1_Start
I don't understand what I'm doing wrong and I've tried using variation of parameters but I end up with a bunch of trig that I can't make anything out of. If someone can point out my error and help with deriving the problem correctly I would really appreciate it.
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