Science:Math Exam Resources/Courses/MATH152/April 2013/Question B 06 (b)
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Question B 06 (b)
Consider the system of linear differential equations given by
Find the general solution of this system of differential equations in real form.
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.
The general real solution of a linear ordinary differential equation is given by
where is one of the eigenvalues of the matrix with appropriate eigenvector .
For our ODE do the following:
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To begin, we compute the eigenvalues of the matrix from part (a) (call this matrix M). To start, compute the characteristic polynomial.
This polynomial has two complex roots given by
Using , we have the eigenvector which can be computed by
Multiplying the first row with 1+i we obtain
The eigenvector is such that . Choosing we have , hence the eigenvector we find is
Important: Please keep in mind that this is one possible eigenvector and there are others that differ from this by a scalar multiple. Since the eigenvalues are complex conjugate, the eigenvector is the complex conjugate of , which is
Now we calculate by using Eulers formula, ,
Hence, the real solution of is given by
Note: For the general (complex) solution you would not drop the part and the solution would be
for constants . But for the real solution, it is enough to only consider .