How to solve linear first order variable coefficient systems?

In summary, the conversation discusses a problem with solving X'(t) = A(t)X(t) using a specific equation and Mathematica code. The issue is that the result obtained is different from that obtained using MATLAB, leading to the realization that the matrix A does not satisfy a certain condition.
  • #1
robijnix
4
0
1. how do i solve
X'(t)=A(t)X(t)

Homework Equations


I know of this equation:
[itex]x(t) = x_0 \exp\left( \int_{t_0}^t f(\xi) \, d\xi \right)[/itex]

but i think that I'm using it wrong.

The Attempt at a Solution


i have the following mathematica code:
A=
fIVwe.png

x = MatrixExp[Integrate[A, {t, 0, s}]].{1, 0,0,0};
Plot[x[[2]] + x[[4]], {s, 0, 500}]

the problem is, that the result i get it not the same as when i numerically plot it with matlab, which leads me to believe that I'm doing something wrong.

any kind of help would be greatly appreciated,

Thanks
 
Last edited:
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  • #2
Never mind i found out the problem.

my matrix A doesn't statisfy A(t1)A(t2)=A(t2)A(t1), in which case the above formula is not valid
 

FAQ: How to solve linear first order variable coefficient systems?

How do you identify a linear first order variable coefficient system?

A linear first order variable coefficient system can be identified by the presence of multiple variables and their coefficients in the system of equations. The variables must also be raised to the first power and there should be no higher order terms.

What is the general method for solving a linear first order variable coefficient system?

The general method for solving a linear first order variable coefficient system is to use the substitution or elimination method. This involves isolating one variable in one equation and substituting its value into the other equations to eliminate it. The resulting equations will have fewer variables and can be solved using basic algebraic techniques.

Can a linear first order variable coefficient system have more than one solution?

Yes, a linear first order variable coefficient system can have multiple solutions. This usually occurs when there are more variables than equations, resulting in an infinite number of solutions. It can also happen when the equations are dependent on each other, resulting in a range of possible solutions.

How do you check if a solution to a linear first order variable coefficient system is correct?

To check if a solution is correct, you can substitute the values of the variables into each equation and see if they satisfy the equations. If all the equations are satisfied, then the solution is correct. You can also graph the equations and see if the solution point lies on all the lines.

Are there any special cases when solving a linear first order variable coefficient system?

Yes, there are two special cases when solving a linear first order variable coefficient system. The first is when the system has no solution, which occurs when the equations are inconsistent and do not intersect. The second is when the system has an infinite number of solutions, which occurs when the equations are dependent on each other and result in the same line.

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