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

 

[blue_raver22]

for your question! The system you have described is known as an ODE (ordinary differential equation) system with variable coefficients. In order to solve this type of system, you will need to use numerical methods such as Euler's method or Runge-Kutta methods. These methods involve approximating the solution at discrete time points and using the values at those points to calculate the solution at the next time point. The accuracy of the solution will depend on the step size chosen for the time points. Additionally, you can also use software packages such as MATLAB or Mathematica to solve this type of system using built-in functions. I hope this helps!