No title

Modeling steps

Define key features of the system of interest: force $F$ , elongation $x$ for elastic bar model
Formulate a mathematical statement relating identified features: $F = k x$ , Hooke's law
Define a model closure: from $n$ observations $(x_{i}, F_{i}), i = 1, \dots, n$ determine $k$ by least squares
Solve the model for cases of interest: for given force $F^{*}$ , the model predicts elongation $x^{*} = F^{*} / k$
Interpret model predictions (solutions): compare predicted $x^{*}$ to observed (experimental) value $x^{ex}$

System	Model	Solution
Falling stone	$y^{''} (x) = g$	$c_{2} x + c_{1} + \frac{g x^{2}}{2}$
Vibrating mass-spring	$k y (x) + m y^{''} (x) = 0$	$c_{2} \sin (\frac{\sqrt{k} x}{\sqrt{m}}) + c_{1} \cos (\frac{\sqrt{k} x}{\sqrt{m}})$
Beam deformation	$EI y^{(4)} (x) = \sin (x)$	$c_{4} x^{3} + c_{3} x^{2} + c_{2} x + c_{1} + \frac{\sin (x)}{EI}$
Parachutist

Table 1. Examples of physical models expressed through ODEs