# Solving System of Differential Equations with Multiple Variables using ode45

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Commented: Star Strider on 9 Feb 2021
Lets say I have a system of six differential equations with multiple variables:
dA = E - C + B - D
dB = F - D + A - C
.....
dF = - B + D - F
How can I write a function which is solvable with ode45? I want a 1x6 initial condition input and a 1x6 output over a time span.

Star Strider on 24 Jun 2020
If you assign ‘A(t)’ as ‘x(1)’ ... ‘F(t)’ as ‘x(6)’, then an anonymous function version would create this matrix:
dfcn = @(t,x) [x(5) - x(3) + x(2) + x(4)
x(6) - x(4) + x(1) - x(3)
. . .
-x(2) + x(4) + x(6)];
and then call ode45 with ‘dfcn’ and the appropriate initial conditions and time span.
The function gets a bit more complicated if involves derivatives of the functions on the right hand side. For that, I usually use the Symbolic Math Toolbox to create the equations, then odeToVectorField to create them as first-order equations, and matlabFunction to convert them to a system that ode45 can use. (Sure, I could do them by hand and then spend a bit of time sorting my algebra errors, but there is no reason to do that.)
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Star Strider on 9 Feb 2021
Anna Jacobsen — Thank you!
Could you clarify what you mean by assigning A(t) to x(1)?
Here is the first equation, ao ‘A(t)’ gets assigned as ‘x(1)’, with the other variables assigned in order, so ‘B(t)’ is ‘x(2)’ and similarly for the others.
Would A(t) be the differential equation dA/dt?
The derivatives are implicitly on the left-hand-side of the equation, so if this were not an anonymous function (and instead a function file), the code for would be:
dxdt(:,1) = x(5) - x(3) + x(2) + x(4);
and similarly for the others.