ODE System with Variable Coefficients

[yashar]

hi
suppose we have this equation :

d/dt(X)=A(t)*X



x is a n by 1 column matrix and A is a n by n matrix that is the matrix of coefficients.
coefficients of equations and consequently A are depend on t which is time.

how i Solve this equation ?

thanks

[lurflurf]

Use the method of peano baker

generalizing separation technique (http://physicsforums.com/showthread.php?t=232776)
Note on the Peano-Baker Method of solving Linear Differential Equations by
Arch Milne (http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=3162564)

[yashar]

hi
in this article that explain the method of peani baker

http://arxiv.org/pdf/1011.1775v1

i can not understand how in page 5 for first example it calculate element (1,2) of I(t)(with subscript n)

can anybody help?

[lurflurf]

by induction
[In(t)](1,2)=∫([A(t)](1,1)[In-1(t)](1,2)+[A(t)](1,2)[In-1(t)](2,2))dt
=∫((1)(tn-1αn-1/(n-1)!)+(t)((a t)n-1/(n-1)!))dt
and so on