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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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# System of 4 linear diffeqs

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