Solving System of ODEs: A Puzzle

AI Thread Summary
To solve the system of ordinary differential equations given by dx/dt = -y and dy/dt = x, one can decouple the equations by substituting one variable into the other. This leads to second-order equations, which can be solved using known functions related to circular and triangular functions. It's noted that some courses may not cover linear systems extensively in a single semester, which can lead to confusion. Extraneous solutions may arise during the solving process, so it's important to substitute back into the original equations to eliminate them. The discussion emphasizes the importance of careful thought and understanding in tackling such problems.
Tony11235
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In order to solve this pde that I'm on, I must solve this system of odes, \frac{dx}{dt} = -y and \frac{dy}{dt} = x , which doesn't look bad, but I haven't had a second semester of ode yet where systems of differential equations are covered. How is this solved?
 
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Tony11235 said:
In order to solve this pde that I'm on, I must solve this system of odes, \frac{dx}{dt} = -y and \frac{dy}{dt} = x , which doesn't look bad, but I haven't had a second semester of ode yet where systems of differential equations are covered. How is this solved?
-The solutions are well known functions.
-There exist EODE class that does not cover linear systems in one semester?
If decoupling the odes is what you whant to do just solve for each variable (trivial) and substitute one into the other.
solve
x=\frac{dy}{dt}
y=-\frac{dx}{dt}
substitute
x=\frac{dy}{dt}=\frac{d}{dt}\left(-\frac{dx}{dt}\right)=-\frac{d^2x}{dt^2}
y=-\frac{dx}{dt}=-\frac{d}{dt}\left(\frac{dy}{dt}\right)=-\frac{d^2y}{dt^2}
Decoupled
Now as for finding soulutions I heard somewhere that some special functions having something to do with cirlces and triangles has something to do with it.
Also be aware you have introduced extraneous solutions so eliminate them be substituting into the original equation.
 
lurflurf said:
-The solutions are well known functions.
-There exist EODE class that does not cover linear systems in one semester?

Ok I think we did. I'm just slow I guess. Maybe I should just not be so quick to ask questions and actually THINK. Sorry I should have recognized the two odes. Simple substitution! I don't know why I was thinking laplace transformations.
 
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