RolyPoly 2 years ago Solve the initial problem \[y_1''=2y_1+y_2+y_1'+y_2'\]\[y_2''=-5y_1+2y_2+5y_1'-y_2'\]\(y_1(0)=y_2(0)=y_1'(0)=4\), \(y_2'(0)=-4\)

1. hba

Integrate.

2. RolyPoly

Note that I put this under linear algebra.

3. RolyPoly

I think it has something to do with eigenvalues and eigenvectors

4. RolyPoly

and diagonalization too

5. experimentX

let y'1 = u, and y'2 = v, you get 4x4 system.

6. RolyPoly

Hmm.. How??

7. experimentX

|dw:1357743467559:dw|

8. experimentX

|dw:1357743488688:dw|

9. experimentX

let that matrix be A, you get X' = AX <-- this is logistic equation ... I must admit ... to me, this is not a nice question.

10. experimentX

the solution is \[ X = Se^{\Lambda t}S^{-1}X(0)\]

11. experimentX

|dw:1357745773804:dw|

12. sirm3d

13. experimentX

|dw:1357746034677:dw||dw:1357746105903:dw|

14. experimentX

|dw:1357746162435:dw|

15. experimentX

|dw:1357746200518:dw||dw:1357746235140:dw|

16. experimentX

so the final solution is |dw:1357746290760:dw|

17. vf321

@sirm3d Using the laplace transform is not as pretty as @experimentX 's cleaner linear algebra solution. You're not guaranteed invertible functions in the frequency domain.