# How do I solve a system of equations?

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Julian Laackmann on 14 Aug 2012
Hello,
suppose I've got four equations which depend on one another and one of them depends on the time:
a = f(b);
b = f(c);
c = f(d);
d = f(a,t);
How can a system like this be solved? I thought about using one of the ode solvers but failed to implement the functions. Can anybody give me a hint?
J

#### 1 Comment

Walter Roberson on 15 Aug 2012
I am confused about you using f() with one argument in most places, but using it with two arguments for "d".

Isktaine on 14 Aug 2012
Edited: Isktaine on 14 Aug 2012
You need to have a function which the ode solvers can act on.
[t,y] = ode45('YourODEFunction', [0 50], [a(0) b(0) c(0) d(0)])
An example of how to create the function:
function dA=YourODEFunction(x,A)
dA(1)=f(b); %Equation for a,
dA(2)=f(c); %Equation for b
dA(3)=f(d); %Equation for c
dA(4)=f(a,t); %Equation for d
dA=dA'
Note that when you have f(b) (and all the others) you'd have to type in an experission eg
dA(1)=3*A(2) %Coding up of a=3*b
Any time your equation would have a 'b' use A(2), any time you would use an 'a' use A(1), any time you would use a 'c' use A(3) and 'd' use A(4). Does that make sense?

Julian Laackmann on 17 Aug 2012
Isktaine,
I've worked with ode solvers before, but I can't get my mind around this one. The results look weird because I know what they should look like (I've implemented the code in Mathematica some time ago).
The issue with the form of the equations for a, b and c just proves that I haven't understood the way the ode solvers are working. If I have an equation that describes a system at a time t I have to hand over the derivative of that equation to the solver, right? Now, if I have some variable a which is indirectly depending on time (since a = f(d(t))) do I have to build the derivative as well? Very confusing, I will have to read the help once more...
Isktaine on 17 Aug 2012
I'm sorry! I think I misunderstood your first question then. I was assuming all of these were differentials i.e. a'=f(b). How silly of me to make that assumption! Are any of the equations actually differentials?
If there are no differentials then you have to uncouple the system before it can solved numerically. You could just use direct substitution to solve them by hand to get one equation for d only in terms of t, the use back substitution to find values for c,b and a once you have d for any t.
Julian Laackmann on 11 Sep 2012
Isktaine,
thank you very much! Uncoupling the equations did the trick.