I have three simultaneous ODEs with coupled dependent variables. I was wondering whether there's a way for Mathematica to decouple these equations (into three ODEs of a single dependent variable each), rather than attempt to solve for the variables (which it currently fails to do).
Specifically, the (messy) equations are $$ \begin{cases} k x_k + k x_A(t) + m \ddot{x}_A(t) = F_D(t) + k x_B(t) \\ 2 k x_B(t) + m \ddot{x}_B(t) = k(x_A(t) + x_C(t)) \\ m \ddot{x}_C(t) = k(x_k + x_B(t) - x_C(t)) \end{cases} $$ where everything that isn't $F_D(t), x_A(t), x_B(t), x_C(t)$ or their derivatives is a constant ($m, k, x_k$). Accompanying is a bunch of initial conditions: $$ x_A(0) = 0, \;x_B(0) = x_k, \;x_C(0) = 2 x_k$$ $$ \dot{x}_A(0) = \dot{x}_B(0) = \dot{x}_C(0) = 0$$ $$ \ddot{x}_A(0) = \ddot{x}_B(0) = \ddot{x}_C(0) = 0$$
From this, I'd like Mathematica to return an equation of the form $$ \ddot{x}_C(t) + \alpha \dot{x}_C(t) + \beta x_C(t) = f(F_D(t)) $$ (I'm only interested in one of the 3 ODEs of this form).
I should note I tried unsuccessfully to use DSolve
, which ran for a very long time then reported For some branches of the general solution, unable to compute the limit
at the given points. Some of the solutions may be lost. and output {}
.
My input was (letting $F_D(t) = F_0 \cos(t \omega)$)
DSolve[{k xk + k xa[t] + m xa''[t] ==
F0 Cos[t \[Omega]] + k xb[t],
2 k xb[t] + m xb''[t] == k (xa[t] + xc[t]),
m xc'')[t] == k (xk + xb[t] - xc[t]), xa'[0] == 0,
xb'[0] == 0, xc'[0] == 0, xa''[0] == 0, xb''[0] == 0,
xc''[0] == 0, xa[0] == 0, xb[0] == xk, xc[0] == 2 xk}, {xa[t],
xb[t], xc[t]}, t].
So, is there anyway to have Mathematica decouple these equations into isolated variables without attempting to solve for the dependent variables? I've heard something vaguely related about using matrices.
Thanks!