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Euge
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Solve the linear system of ODE ##x' = 2x + 3y##, ##y' = -3x + y## with initial conditions ##x(0) = 1, y(0) = 2##.
julian said:By the way, there are other methods of solving the problem!
Yes, they are functions of ##t##.Mayhem said:When working with these types of problems is it implied that x = x(t) and y = y(t).
Because you have a second order differential equation the general solution will involve two unknown constants, so you need ##x(0)## and ##x'(0)##. Now, ##x'(0)## was not specified in the question but that does not present a problem because you are given a formula for ##x'(t)##. Yep?erobz said:The system is equivalent to two second order ODE's
$$ y'' - 3y'+11y , y(0)=2$$
$$ x''-3x'+11x=0, x(0)=1 $$
Algebraic error fixed ( hopefully)...
Using the original equations and given initial conditions we find ## x'(0) = 8, y'(0) = -1##julian said:Because you have a second order differential equation the general solution will involve two unknown constants, so you need ##x(0)## and ##x'(0)##. Now, ##x'(0)## was not specified in the question but that does not present a problem because you are given a formula for ##x'(t)##. Yep?
julian said:Do you know how to go about solving your differential equation? It is a homogeneous linear differential equation of second order with constant coefficients.
But different constants in each case, not the same.erobz said:Using the original equations and given initial conditions we find ## x'(0) = 8, y'(0) = -1##
With an old textbook in front of me... Find the auxiliary equation:
$$m^2 -3m+11 = 0 \implies m = \frac{3}{2} \pm \frac{\sqrt{35}i}{2} $$
I think that implies the general solution ( for ##y## or ##x## - same ODE - different i.c. ):
$$ x(t) = y(t) = e^{ \left( \frac{3}{2}t \right) } \left( c_1 \cos \left( \frac{ \sqrt{35} }{2} t \right) + c_2 \sin \left( \frac{ \sqrt{35} }{2} t \right) \right) $$
Yeah, I was just being lazy. Thats why I said the general solution is... the constants would be determined for the initial conditions of each. I would think those constants aren't necessarily distinct in the general solution...so I thought I could get away with it?bob012345 said:But different constants in each case, not the same.