System of 4 linear diffeqs

In summary, you need to provide constraints for the initial conditions or the function to solve the system.
  • #1
169
0
Hi!

I have a system of four differential equations:

[itex]\dot{x}(t)=Ax(t)+a[/itex]

where [itex]A\in R^{4x4}[/itex] and [itex] a\in R^{4}[/itex] are known and, [itex]x(t)\in R^4 \forall t>0[/itex]

EDIT: the constraints I have on these differential equations are: [itex]x_1(0)=x_{1,0}, x_2(0)=x_{2,0}, x_3(T)=x_{3,T}, x_4(T)=x_{4,T}[/itex]

I know I can decompose [itex]A=MDM^{-1}[/itex], where [itex]D=diag(\lambda_1, ... , \lambda_4)[/itex], and M is the matrix of eigenvectors (all of which I have computed the exact numbers of, important may or may not be that Re(lambda_i)>0 for i=1,2 and Re(lambda_i)<0 for i=3,4). By defining [itex]y=M^{-1}x[/itex], I can rewrite this system into:

[itex]\dot{y}(t)=Dy+M^{-1}a[/itex]

Which I could solve had I known the initial conditions y_i(0) or the boundary conditions y_i(T). But I don't. I know (as this has been done before in a paper that does not elicit these technicalities) that I should somehow find y_i (0) or y_i (T) for each i=1,...,4. But how?
 
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  • #2
Hey jacobrhcp.

You need to provide some kind of constraint for the function or the initial condition to obtain the complete explicit definition of the system.

Even if you have to infer it in some way, it needs to be done if you want to calculate stuff and get numeric answers.

If this is an assignment then you should talk to your lecturer. If it's for a real world problem, then you need to think about any information that you can obtain that helps lead to using a reliable assumption that can give you a value or at least constrain it to a point where something reasonable can be used.
 
  • #3
Yes! of course, you are very right, I am very sorry I did not write them down in the initial problem description, I've edited them in. Without these it's all nonsense. Though it's no assignment, it is a purely mathematical problem.

(in fact if you're interested: it is was mathematical model of the economy, that I linearized into a system like described above, and now I know it can be solved as some other people did it in papers and mumble words about linearization and then transformations, but I'm stuck on how to check and redo their work)
 

1. What is a system of 4 linear differential equations?

A system of 4 linear differential equations is a set of four equations that describe the change of multiple dependent variables with respect to one or more independent variables. These equations are linear, meaning that the variables are only raised to the first power and there are no products or compositions of the variables.

2. How is a system of 4 linear differential equations solved?

A system of 4 linear differential equations can be solved using various methods, such as substitution, elimination, or using matrices. The most common method is to use matrix operations to solve for the values of the dependent variables.

3. What is the importance of solving a system of 4 linear differential equations?

A system of 4 linear differential equations is important in many areas of science, such as physics, engineering, and economics. It allows us to model and predict the behavior of complex systems by understanding how the dependent variables change with respect to the independent variables.

4. Can a system of 4 linear differential equations have an infinite number of solutions?

No, a system of 4 linear differential equations can only have a unique solution or no solution at all. This is because the equations are linear and can be solved using matrix operations, which always result in a unique solution.

5. What are some real-world applications of a system of 4 linear differential equations?

A system of 4 linear differential equations is used in various real-world applications, such as modeling population growth, analyzing chemical reactions, and predicting the behavior of electronic circuits. It is also used in control systems, robotics, and weather forecasting.

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