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Matrix ODEby Manchot
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#1
Jan1514, 01:03 PM

P: 728

I'm trying to find a general solution for the logistic ODE [itex]\frac{dU}{dx}=A(IU)U[/itex], where A and U are square matrices and x is a scalar parameter. Inspired by the scalar equivalent I guessed that [itex]U=(I+e^{Ax})^{1}[/itex] is a valid solution; however, [itex]U=(I+e^{Ax+B})^{1}[/itex] is not when U and A don't commute. Any ideas?



#2
Jan1514, 02:33 PM

PF Gold
P: 377

The general solution to the scaler equation is:
E^(A x)/(E^(A x) + E^C) where C is a constant. Maybe this can lead to a similar solution for the matricial version? 


#3
Jan1514, 02:46 PM

PF Gold
P: 377

If U is a function of A,
then U commutes with A. 


#4
Jan1514, 06:21 PM

P: 728

Matrix ODE
I tried all sorts of versions of the scalar equation, maajdl. They all run into the same commutation problem. Unfortunately, U is a function of both A and the initial condition, which means that it doesn't commute with A unless the initial condition does.



#5
Jan1614, 01:09 AM

PF Gold
P: 377

Could that help?
Assuming: A = M^{1}DM where D is a diagonal matrix V = MUM^{1} The ODE becomes: dV/dx = D(IV)V 


#6
Jan1614, 06:40 AM

P: 728

Yeah, I tried diagonalizing both A and the initial condition. No dice.



#7
Jan1614, 07:48 AM

PF Gold
P: 377

What is your practical goal?
Why do you need a formal solution? 


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