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
Note on the Peano-Baker Method of solving Linear Differential Equations by
Arch Milne

 

[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
[I_n(t)]_(1,2)=∫([A(t)]_(1,1)[I_n-1(t)]_(1,2)+[A(t)]_(1,2)[I_n-1(t)]_(2,2))dt
=∫((1)(t^n-1α_n-1/(n-1)!)+(t)((a t)^n-1/(n-1)!))dt
and so on