Use the eigenvalue method to find a general solution of the following system:?

Question

x'1 = x1 - 5x2
x'2 = x1 - x2

i have found the eigenvalue to be complex... + or - 2i

can anyone elaborate on how I go on to find the general solution from here???

btw 1 or 2 is not powers...

Answer 1

x'=Ax
A=[1, -5]
.... [1 -1]

det (A-λI) = (1-λ)(-1-λ) + 5
-(1-λ²) + 5=0
λ²+4=0
λ=±2i

Now, we find an eigenvector of the system:

ker [1-2i, -5] = span (transpose([1+2i, 1]))
..... [1, -1-2i]

So one of the complex solutions of this system is:

[(1+2i) e^(2it)]
[e^(2it)]

Writing this as the sum of real and imaginary parts:

[(1+2i)(cos (2t) + i sin (2t))]
[(cos (2t) + i sin (2t))]

[cos (2t) - 2 sin (2t)] + i [sin (2t) + 2 cos (2t)]
[cos (2t)] ...................... [sin (2t)]

And the real and imaginary parts together form a fundamental set of solutions. Therefore, your general solution is au+bv, where a and b are scalars, u=transpose ([cos (2t) - 2 sin (2t), cos (2t)]), and v=transpose ([sin (2t) + 2 cos (2t), sin (2t])

Answer 2

You still need to find eigenvectors for each eigenvalue.
So suppose [a1 b1] is an eigenvector for +2i and [a2 b2] is one for -2i. Then the general solution will be
x1=a1*e^(2it) +a2*e^(-2it)
x2=a2*e^(2it)+b2*e^(-2it)

In general, the a's and b' s will be complex, but you can use Euler's formula to convert these answers to sums of sines and cosines.

Answer 3

i am going to circumvent ahead and use lambda, pondering the actual undeniable actuality that your e book likely does: y" + ?y = 0 has function equation: r² + ? = 0 --> r = ± i??. So the eigenvalues of the equation are r = ± i??. The eigenfunctions, then, are given with the help of ability of e^(ix??) and e^(-ix??). those verify that you do searching that the eigenfunctions are really sin(x??) and cos(x??).