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3 Pages

### hw9-6

Course: MATH 308, Fall 2008
School: Texas A&M
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Word Count: 417

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to Solutions Homework from Section 9.6 Problem 3 The matrix has eigenvector/eigenvalue pairs equal to 1 0 1 = 1, u1 = 0 -1 - 2i 1 2 = 1 + i, u2 = i -1 + 2i 1 3 = 1 - i, u3 = -i The general solution is therefore 1 -1 - 2i -1 + 2i + c3 e(1-i)t 1 1 c1 et 0 + c2 e(1+i)t 0 i -i We can use Euler's formula, eit = cos(t) + i sin(t), to expand this to -1 - 2i -1 + 2i 1 +c3 et (cos(t)-i sin(t))...

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to Solutions Homework from Section 9.6 Problem 3 The matrix has eigenvector/eigenvalue pairs equal to 1 0 1 = 1, u1 = 0 -1 - 2i 1 2 = 1 + i, u2 = i -1 + 2i 1 3 = 1 - i, u3 = -i The general solution is therefore 1 -1 - 2i -1 + 2i + c3 e(1-i)t 1 1 c1 et 0 + c2 e(1+i)t 0 i -i We can use Euler's formula, eit = cos(t) + i sin(t), to expand this to -1 - 2i -1 + 2i 1 +c3 et (cos(t)-i sin(t)) 1 1 c1 et 0 +c2 et (cos(t)+i sin(t)) i -i 0 We can then take the Real part of this to get a real valued solution: Problem 5 The eigenvalue/eigenvector pairs are 1 = -1 + 4i, u1 = 1 = -1 - 4i, u2 = 1 i/2 1 -i/2 1 , , The general solutions are therefore x1 (t) = e(-1+4i)t x2 (t) = e(-1-4i)t The real part is given by Re(x1 (t)) = Re(x2 (t)) = e-t - sin(4t)/2 cos(4t) i/2 1 -i/2 1 = e-t (cos(4t) + i sin(4t)) = e-t (cos(4t) - i sin(4t)) i/2 1 -i/2 1 The imaginary part by Im(x1 (t)) = Im(x2 (t)) = e-t cos(4t)/2 sin(4t) The Real and Imagingary parts, taken separately, linearly are independent. Therefore the fundamental matrix is given by e-t - sin(4t)/2 cos(4t)/2 cos(4t) sin(4t) Note: The solution in the book is found by switching the first and second column, and multiplying each column by an appropriate constant. Problem 13 The fundamental solutions are given by x1 (t) = e(-2+i)t x2 (t) = e(-2-i)t Taking the Real and Imaginary part of x1 (t) = e-2t (cos(t) + i sin(t)) 2 1 -1 - i 1 -1 - i 1 -1 + i we get Re(x1 (t)) = e-2t cos(t) - cos(t) + sin(t) sin(t) - cos(t) - sin(t) Im(x1 (t)) = e-2t So we can take the fundamental matrix to be ...

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