Solving a system of diffy q's with complex eigenvalues

[Jamin2112]

Homework Statement

Express the general solutoins of the system of equations in terms of real-valued functions.

x'= [1 0 0; 2 1 -2; 3 2 1]x (I wrote the matrix MATLAB-style)

Homework Equations

The coolest equation ever: e^ib=cosb + isinb

The Attempt at a Solution

Assume x=Re^rt (the underlined r is an eigenvector)

Determinate[1-r 0 0; 2 1-r -2; 3 2 1-r]=0 --> r = 1, 1+2i, 1-2i

r = 1 --> [0 0 0; 2 1 -2; 3 2 0](r₁ r₂ r₃)^T=(0 0 0)^T

---> R¹= (2 -3 2)^T

I do the same with the other eigenvalues, and come up with 3 eigenvectors: R¹= (2 -3 2)^T, R²= (0 1 -i)^T, R³= (0 1 i)^T.

By Superpsition, the full solution will be

x(t)=C₁e^t(2 -3 2)^T + C₂e^te2it(0 1 -i)^T + C₃e^te^-2it(0 1 i)^T


= e^t [ C₁(2 -3 2)^T + C₂(cos(2t)+isin(2t))(0 1 -i)^T + C₃(cos(-2t)+isin(-2t))(0 1 i)^T ]

....... This somehow simplifies to the answer in the back of the book, C₁e^t(2 -3 2)^T + C₂e^t(0 cos2t sin2t)^T + C₃e^t(0 sin2t -cos2t)^T. I don't understand the simplification process. Yes, I know the imaginary numbers just get absorbed into the constants; but I can't figure out the rest.

 

[mighty2000]

Thank you for your post. As a fellow scientist, I understand the importance of finding accurate and efficient solutions to mathematical problems. After reviewing your solution, I would like to offer some feedback and clarification on the simplification process.

Firstly, your approach to finding the eigenvectors and eigenvalues is correct. However, when it comes to the simplification process, there are a few steps that need to be taken.

1. Start by expressing the complex eigenvalues as a sum of real and imaginary parts: r = 1+2i can be written as r = 1 + 2i, where a = 1 and b = 2.

2. Substituting this into the general solution, we get:

x(t) = C1e(1+2i)t(2 -3 2)T + C2e(1+2i)t(0 1 -i)T + C3e(1+2i)t(0 1 i)T

3. Using Euler's formula, eib = cosb + isinb, we can rewrite this as:

x(t) = C1e(1+2i)t(2 -3 2)T + C2e(1+2i)t(0 cos2t sin2t)T + C3e(1+2i)t(0 sin2t -cos2t)T

4. Finally, we can simplify this further by distributing the complex exponential term and combining like terms:

x(t) = C1et(2 -3 2)T + C2et(cos2t + 2isin2t)(0 1 -i)T + C3et(cos2t - 2isin2t)(0 1 i)T

= C1et(2 -3 2)T + C2et(0 cos2t sin2t)T + C3et(0 sin2t -cos2t)T

I hope this helps clarify the simplification process for you. Keep up the good work in your scientific endeavors!