ODE Models: 1.3

Almost everything in Science and Engineering is described by a DE of some sort.

TabView[{
"Pop"->TabView[{
"Words"->"Population growth is proportional to population","Eqs"->"(dP/dt) = k P"
}],
"Decay"->TabView[{
"Words"->"Decay rate is proportional to the amount","Eqs"->"(dA/dt) = k A"
}],
"Cooling"->TabView[{
"Words"->"Rate of change of temperature is proportional to difference from surrounding temperature","Eqs"->"(dT/dt) = k (T-(Subscript[T, m]))"
}],
"Disease"->TabView[{
"Words"->"Rate of infection is proportional to number of susceptibles and number of infected","Eqs"->"(di/dt) = k i s = k i (n - i)"
}],
"Mix"->TabView[{
"Pic"->,
"Words"->"Rate of change of salt = rate in - rate out","Eqs"->"(dS/dt) = (Subscript[gpm, in]) (Subscript[c, in]) - (Subscript[gpm, out]) (Subscript[c, out])"
}],
"Tank"->TabView[{
"Pic"->,
"Words"->"exit velocity equals drop velocity","Preliminary Eq"->"\!\(\*Cell[TextData[Cell[BoxData[
FormBox[
RowBox[{"v", "=", 
SqrtBox[
RowBox[{"2", " ", "g", " ", "h"}]]}], TraditionalForm]],
FormatType->"TraditionalForm"]]]\)",
"ODE"->"\!\(\*Cell[TextData[Cell[BoxData[
FormBox[
RowBox[{
FractionBox["dh", "dt"], "=", 
RowBox[{"-", 
FractionBox[
RowBox[{
SubscriptBox["A", "h"], 
SqrtBox[
RowBox[{"2", " ", "g", " ", "h"}]]}], 
SubscriptBox["A", "w"]]}]}], TraditionalForm]],
FormatType->"TraditionalForm"]]]\)"
}],
"Circuit"->TabView[{
"Pic"-> ,
"i and q"->"q is capacitor charge i=dq/dt",
"Components"->TabView[{
"L"->,
"R"->,
"C"->
}],
"Words"->"Voltage round loop adds to zero","Eqs"->"L q'' + R q' + 1/C q = E(t)"
}],
"Falling"->TabView[{
"Pic"-> ,e
"Words"->"F = m a",
"Eqs"->"m s'' = - m g",
"Sol"-> "-0.5 g (t^2)+(Subscript[v, 0])t + (Subscript[s, 0])"
}]
}
]

Usefull Commands

DE

q[t]/.DSolve[
0.001 q''[t]+ 0.001 q'[t]+ 0.1 q[t]==0.1,
q[t],
t][[1]]

Linear Algebra

A=(({{1, 4}, {1, -2}}));
Eigenvalues[A]
Eigensystem[A]