MHB Direct solution for two unknowns in two equations

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The discussion centers on solving two equations with two unknowns, specifically focusing on the equations involving coefficients C1, C2, C3, and known values Q1 and Q2. The original poster has attempted substitution and elimination methods but finds them leading to complex quartic expressions. They express a preference for iterative methods due to their stability and quick convergence, despite acknowledging the existence of a direct solution method for quartics. Another participant agrees that the iterative approach is simpler and more practical for this problem. The consensus leans towards using approximation methods for ease of computation.
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I am writing a computer program to solve a physical problem which at some part involves the following two equations and two unknowns;

(1) \[ C_1(x^4-y^4) + C_2x = Q_1 \]
(2) \[ C_1(y^4-x^4) + C_3y = Q_2 \]

C1, C2 & C3 are coefficients which can readily be calculated and do not rely on knowing the values of X or Y. Q1 and Q2 are known values (effectively boundary conditions). I've tried substitution and elimination but end up with a much more complicated expression which looks to be a quartic equation. I know I can solve this iteratively and it does converge quickly and is stable. However, is there any method that can solve it directly and thus avoid the iterative approach?
 
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I see no way to avoid the quartic. There is a method to do this to get an exact answer (here) but I'd stick with approximation. It's much easier to work out.

-Dan
 
topsquark said:
I see no way to avoid the quartic. There is a method to do this to get an exact answer (here) but I'd stick with approximation. It's much easier to work out.

-Dan
Hi Dan, I took a look at that solution for the quartic and you're not wrong the iterative method (i.e. approximation) is far easier to work out!

Ian
 
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