# Question:How to solve the Second Order ODE System?

## Question:How to solve the Second Order ODE System?

Maple

How to solve the following second order ODE system? (I want to solve analytically and numerically and to compare the results) Thanks. Best regards.

where

Initial conditions:

Other symbols in the equation (v_0,a_0 etc.) are constant.

MY CODE TRY:  Code.mw

I tried to write a code for matrices in the question. ( I used x instead of gamma in the code)

```restart:
n:=2:
M:= tau -> Matrix
( n,
n,
shape=identity
)
+Matrix
( n,
n,             (i,j)->2*x*sin(i*Pi*v_0*tau)*sin(j*Pi*v_0*tau)
):
C:=tau -> Matrix
( n,
n,
(i,j)->4*(i*Pi*v_0*tau)*sin(i*Pi*v_0*tau)*cos(j*Pi*v_0*tau)
):
K:=tau -> n^4*Pi^2* Matrix
( n,
n,
shape=identity
)-Matrix
( n,
n,
(i,j)-> 2*mu*(i*Pi*v_0)^2*sin(i*Pi*v_0*tau)*sin(j*Pi*v_0*tau)+2*mu*(i*Pi*a_0)*sin(i*Pi*v_0*tau)*cos(j*Pi*v_0*tau)
):
f:=tau -> Vector
( n,
(i,j)-> x*Pi^2*(Pi/2+v_0^2*sum(((1-(-1)^k)/k^3)*sin(k*Pi*v_0*tau),k=1..infinity)-a_0*sum(((1-(-1)^k)/k^4)*cos(k*Pi*v_0*tau),k=1..infinity))*sin(i*Pi*v_0*tau)
):

X:= Vector(n, i-> x[i](t)):
sys:= M(tau).diff~(X, tau\$2)+C(tau).diff~(X, tau)+K(tau).X=f(tau):

```

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