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.