I have a equation of motion for a forced pendulum show below $$ {d^2\theta\over dt^2} = -{g\over L}\sin\theta + C\cos\theta\sin(Dt) $$ $L=10$ cm, $C=2\ \hbox{s}^{-2}$ and $D=5\ \hbox{s}^{-1}$.
I am trying to make this equation dimensionless by setting the follow equations
$$\omega^2 = g/L,\quad \beta=D/\omega,\quad \gamma=C/\omega^2$$
and changing the variable $x = \omega t$.
However I can't get rid of all dimensions. Can I have help making this question dimensionless ?
Introduce $\lambda=L/L_C$ and $\tau=t/t_C$. Then the equation reads
$$\frac{1}{t_C^2} \frac{d^2 \theta}{d \tau^2} = -\frac{g}{L_C \lambda} \sin(\theta) + C \cos(\theta) \sin(D t_C \tau).$$
Clear the factor of $t_C^2$:
$$\frac{d^2 \theta}{d \tau^2} = -\frac{g t_C^2}{L_C \lambda} \sin(\theta) + C t_C^2 \cos(\theta) \sin(D t_C \tau).$$
You have three parameter aggregates and two parameters to play with, so only two of the parameter aggregates can be made to be $1$. For instance you can take $L_C=\frac{1}{g t_C^2}$ and $t_C=C^{-1/2}$ to get
$$\frac{d^2 \theta}{d \tau^2} = -\frac{1}{\lambda} \sin(\theta) + \cos(\theta) \sin(D C^{-1/2} \tau).$$
In this formulation, $DC^{-1/2}$ is a dimensionless parameter. You could instead decide $t_C=D^{-1}$, in which case you get
$$\frac{d^2 \theta}{d \tau^2} = -\frac{1}{\lambda} \sin(\theta) + \frac{C}{D^2} \cos(\theta) \sin(\tau).$$
where again $\frac{C}{D^2}$ is a dimensionless parameter.
Sometimes in modeling, we want our dimensionless parameters to be small, so that we can do some sort of perturbative approximation. To this end, you would choose the first one if $D$ were small, meaning that the oscillation of the forcing is slow. You would instead choose the second one if $C$ were small, meaning that the strength of the forcing is small. With your numbers, the second one would be useful.