Application: Falling Object

DE in v(t)

Here we will use command DSolve to solve the DE.

Clear[y]
deq=m*v'[t]==m*9.8-γ*v[t]
DSolve[deq,v[t],t]

Here is a plot of the familty of curves that satisfy the DE:

family=Flatten[((49 m)/(5 γ))+(E^(-((t γ)/m))) C[1]/.C[1]->{-20,-10,0,10,20}]

To get a plot of the family of curves, I need to specify values for the unspecified constants m and γ.

velocityplot=Plot[Evaluate[family/.m->1/.γ->1/2],{t,0,10},PlotStyle->Thick,AxesLabel->{"Time","Velocity"},BaseStyle->Large]

The terminal velocity of all trajectories is ((49 m)/(5 γ)), since the exponential part of the solution approaches zero as t approaches infinity.

We shall see that this DE can be solved by hand using the integrating factor technique, since it is 1st order, constant coefficient and linear in the unknown function v(t).

DE in x(t)

Here is the same physical system modeled by a 2nd order, constant coefficient linear DE:

Clear[y]
deq=m*x''[t]==m*9.8-γ*x'[t]
DSolve[deq,x[t],t]

Now, the family of curves is doubly infinite since we have two constants in our solution.

family=Flatten[((49 m t)/(5 γ))-(((E^(-((t γ)/m))) m C[1])/γ)+C[2]/.C[1]->{-20,0,10,20}/.C[2]->{-25,0,5,25}]

Plot[Evaluate[family/.m->1/.γ->1/2],{t,0,10},PlotStyle->Thick,AxesLabel->{"Time","Position"},BaseStyle->Large]

The particle approaches a terminal velocity of  ((49 m)/(5 γ)) and the position approaches a linear relationship (since velocity is approaching a constant value).

Notice the particular linear relationship depends on the values of the parameters in the model.

DE in x(t), v(t)

Here is the same physical system modeled by a 1st order, constant coefficient linear system of DEs:

Clear[y]
m=1
γ=1/2
deq1=m*v'[t]==m*9.8-γ*v[t]
deq2=x'[t]==v[t]
DSolve[{deq1,deq2},{x[t],v[t]},t];
Simplify[%]

Now, the family of curves is doubly infinite since we have two constants in our solution.

family=Flatten[-39.2+19.6 t+(2-2(E^(-0.5 t))) C[1]+1.C[2]/.C[1]->{-20,0,10,20}/.C[2]->{-25,0,5,25}]

Plot[Evaluate[family/.m->1/.γ->1/2],{t,0,10},PlotStyle->Thick,AxesLabel->{"Time","Position"},BaseStyle->Large]

The particle approaches a terminal velocity of  ((49 m)/(5 γ)) and the position approaches a linear relationship (since velocity is approaching a constant value).

Notice the constants C[1] and C[2] do not give the same members of the family of curves that we saw before!

Direction Fields

A direction field shows the global behaviour of the solution to the DE, and must be constructed using a computer. It is important to note that direction fields can be constructed without knowing the solution to the DE. The slopes of the arrows represent the tangent to the solution to the DE at the point.

deq=y'[t]==t-y[t]
field=VectorPlot[{1,t-y},{t,-10,20},{y,-10,20},VectorScale->{Tiny,Small,None},FrameLabel->{"t","y"},BaseStyle->Large]

Here is another example.

deq=y'[t]==y[t] (t^2) Sin[y[t]]
field=VectorPlot[{1,y (t^2) Sin[y]},{t,-3,3},{y,-3,3},VectorScale->{Tiny,Small,None},FrameLabel->{"t","y[t]"},BaseStyle->20]

If you can find a solution to the DE, you can overlay it on the direction field using the Show command.

Direction Fields for the velocity of our falling object

The terminal velocity of all trajectories is ((49 m)/(5 γ)). Our direction field should show an equilibrium solution that increases in value as m increases, and decreases in value as γ increases.

Clear[field]
field[m_,γ_]:=VectorPlot[{1,98/10-γ*v/m},{t,0,50},{v,-10,50},VectorScale->{Tiny,Small,None}];

Manipulate[Show[field[m,γ],Frame->True,FrameLabel->{"t","velocity"},ImageSize->{500,500},PlotRange->{{0,50},{-10,50}},BaseStyle->Large],{m,1,10},{γ,0,2}]

Initial Value Problems

Here, we solve a IVP by including the initial condition in the DSolve command.

Clear[y]
deq= y'[x] ==Cos[x]
DSolve[deq,y[x],x]
DSolve[{deq,y[0]==2},y[x],x]

Notice the second DSolve included an initial condition and automatically found the solution to the IVP.

f[x_]=Sin[x]+C;
list=Table[f[x]/.C->i,{i,-11,10,3}];
plot2=Plot[Evaluate[list],{x,-4,4}];

plot1=Plot[Sin[x]+2,{x,-4,4},PlotStyle->{Red,Thick}];
field=VectorPlot[{1,Cos[x]},{x,-4,4},{y,-13,13},VectorScale->{Tiny,Tiny,None},VectorPoints->10];

Show[field,plot1,plot2,Graphics[Disk[{0,2},{0.2,0.6}]],Frame->True,FrameLabel->{"t","y"},ImageSize->{500,500},AspectRatio->1]