i have the following ODE of a single mass oscillator with a spring and a damper:
$m*\ddot{s}=-d*\dot{s}-c*s+m*g$
where $m = 5$, $g=10$
$c$ & $d$ are variable
The ODE is now :
$\ddot{s}=-\frac{d}{5}*\dot{s}-\frac{c}{5}*s+10$
I have 2 intitial conditions that have to apply:
$s(0)=0$
$\dot{s}(0)=0$
Now to my question: If I calculate the solution of the oscillation (e.g. with Matlab) I get
$s(t,c,d)$ which is still depending on the variables $c$ an $d$
Further i can calculate the derivatives of the solution depending on $c$ and $d$
$\dot{s}(t,c,d)$
$\ddot{s}(t,c,d)$
My problem is, that i have 3 conditions for 2 unknown variables, which makes the linear system of equations needed to solve for $c$ and $d$ overdetermined.
$s(t_{max})=s_{max}$
$\dot{s}(\ddot{s}=0)=\dot{s}_{max}$
$\dot{s}(t_{max})=0$
How do I solve for $c$ and $d$ so that all the conditions including the initial conditions are met and valid for this ODE?