Make change of variable by putting "xdot=y and ydot=x twodots". After can study the equilibrium and the phase portait of the system. Thanks
here is the solution (see attached) http://www.wolframalpha.com/input/?i=x%27%27+-+2x%27+%2B+2x+%3D+0
There is standard theory to solve the problem the way people learn in ordinary differential équations for mathematicians. See comments above. There is also a physical way to solve it without much calculations.
Observe that the differential equation is linear in x and has constant coefficients. This qualifies the constant solution zero as the only stationary point we need to assess. We can observe that the equation models a harmonic oscillator with negative damping. From the physical perspective this means that we inject more and more energy into this system. Indeed, upon setting x(t) =exp(t) *u(t), we note that u(t) =a*sin(t) +b*sin(t) with a and b reals. Since the energy of the harmonic oscillator is increasing as exp(2*t) over time as is seen from the solution above, we see that the stationary point must be unstable. This corresponds to the criterion for the real part of the associated first order system be positive that the OP certainly has heard of.
The code for plotting has been posted before.
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