Systems of ODE: Converting complex solution to real

Shaybay92
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Homework Statement



So, I have found a general solution to a system of linear first order ODE's and this is what I got:

X = c1v1e^(-1+2i)t + c2v2e^(-1-2i)t

where v1 = [-1+2i, 5], v2=[-1-2i,5]. The question is, how do I now change this solution into its real equivalent? i.e. I don't want any complex numbers in my solution.

I have a textbook which explains but it still doesn't make sense to me how they manage to go from a complex to real solution. Could someone explain step by step? Thanks.
 
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I think they're saying that X=x+yi, just resolve in real and imaginary parts...
 
I don't think so, because it has to still span the solution space, and merely dropping the imaginary parts will not ensure this.
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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