ODE Linear System Complex Eigenvalues

[Lahooty]

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.

 

[STEMucator]

So you have a non homogeneous system you want to solve. Let's denote the system x' = Ax + g(t)

The first thing you should do is solve the homogeneous system x' = Ax by solving the characteristic polynomial p_A(λ) for your eigenvalues and then proceed to find your eigenvectors. The eigenvectors will be easy since one is just a complex conjugate of the other.

These will give you your fundamental homogeneous solution, let's call it x_c.

Then you can construct a fundamental matrix ψ(t) from the columns of your homogeneous solution.

Then to solve the non homogeneous system for a particular solution x_p, I would recommend variation of parameters. So you should assume your particular solution has the form : x_p = ψ(t)u(t) where u(t) satisfies g(t) = ψ(t)u'(t).

After you solve g(t) = ψ(t)u'(t) for u_{1}^{'} and u_{2}^{'}, integrate them to find u_1 and u_2 which finally give you your vector u(t). Then simply do some matrix multiplication to find x_p.

After solving for the homogeneous solution and the particular solution, your general solution will be x = x_c + x_p

I hope this helps you. It's a lot of work, but it's doable.