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Can a system of differential equations have a solution when the rate of change of a variable is constant? And how to calculate it?

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There is a system of differential equations containing more than two variables, and the rate of change (with time) of one of the variables is constant c as shown in the following. That is, the differentials of the variables cannot all be zero at the same time, and the equilibrium point may not exist. Can one still calculate this system. Thank you.

Accepted Answer

Walter Roberson
Walter Roberson on 23 Aug 2021
There is no problem having a rate of change that is a constant. For example if you were using an ode you might have
ode45(@(t,x) [x(1)+x(2)^2, -1.234])
and that would be for the case where x'' is the constant -1.234
Your system is a PDE rather than an ODE, but Yes, you can do it.
If you are constructing symbolic expressions, then make sure that dz is a symbolic function, such as
syms x y z
dz(x,y,z) = c;
do not use
dz = c;
in this case.
The reason for this is that in the case you just assign a numeric constant, then matlabFunction(dz) would generate @() c -- a function with no parameters permitted. But if you define dz(x,y,z) = c then matlabFunction(dz) would generate @(x,y,z) c -- which permits (but ignores) inputs.
Walter Roberson
Walter Roberson on 24 Aug 2021
The "TL;DR" would be that the meaning of notations is different in different fields and at different times, and it is very often a mistake to say that a particular notation only means one particular thing.
The task of reconstructing an equation from its multiple derivatives was addressed in the Partial Differential Equations sections of my courses.

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