Finite difference discretization for systems of higher ODEs

[Ojo Saheed]

How can I use finite difference to discretize a system of fourth order differential equations?
for example:
y⁽⁴⁾+5y⁽³⁾-2y^''+3y^'-y=0

 

[Chestermiller]

This doesn't look like a system of 4th order ODEs. It looks like a single 4th order ODE.

Chet

 

[Ojo Saheed]

This doesn't look like a system of 4th order ODEs. It looks like a single 4th order ODE.

Chet

Thanks for the reply Chet. Actually, I had thought that given an example of one the system of equations might suffice for a demonstration. In any case, a better representation of the system goes thus:
y₁⁽⁴⁾=a₁y₁^''+a₂y₂^''+a₃y₃^''+a₄y₁+a₅y₂+a₆y₃+a₇
y₂⁽⁴⁾=b₁y₁^''+b₂y₂^''+b₃y₃^''+b₄y₁+b₅y₂+b₆y₃+b₇
y₃⁽⁴⁾=c₁y₁^''+c₂y₂^''+c₃y₃^''+c₄y₁+c₅y₂+c₆y₃+c₇
where a, b and c terms are constant coefficients

 

[Chestermiller]

Thanks for the reply Chet. Actually, I had thought that given an example of one the system of equations might suffice for a demonstration. In any case, a better representation of the system goes thus:
y₁⁽⁴⁾=a₁y₁^''+a₂y₂^''+a₃y₃^''+a₄y₁+a₅y₂+a₆y₃+a₇
y₂⁽⁴⁾=b₁y₁^''+b₂y₂^''+b₃y₃^''+b₄y₁+b₅y₂+b₆y₃+b₇
y₃⁽⁴⁾=c₁y₁^''+c₂y₂^''+c₃y₃^''+c₄y₁+c₅y₂+c₆y₃+c₇
where a, b and c terms are constant coefficients

If you're trying to solve this numerically, one way it is often done is to convert to a set of first order ODEs. In this particular case, there would be 12 first order ODEs to integrate.

Define,
z_j=y_j (j = 1,3)
z_j+3=y'_j (j = 1,3)
z_j+6=y''_j (j=1,3)
z_j+9=y'''_j (j=1,3)
Then
z₁' = z₄
z₄' = z₇
z₇' = z₁₀
z₁₀' = a₁z₇+a₂z₈+a₃z₉+a₄z₁+a₅z₂+a₆z₃+a₇
etc.

You end up with 12 coupled first order linear ODEs explicit in the derivatives. You can use an automatic integrator to solve them or use your own coding of Runge Kutta or forward Euler, or backward Euler, or whatever.

Chet