6701 ODE Notes/
Final Exam 2020

Plotting a vector field

VectorPlot[{(1/2)  x (1-x/5-y),(1/10) y (-1+2 x-3 y/10)},{x,0,2},{y,0,2}]

interest=VectorPlot[{(1/2)  x (1-x/5-y),(1/10) y (-1+2 x-3 y/10)}/Norm[{(1/2)  x (1-x/5-y),(1/10) y (-1+2 x-3 y/10)}],{x,0,2},{y,0,2}]

Equilibrium Point

field[x_,y_]={(1/2)  x (1-x/5-y),(1/10) y (-1+2 x-3 y/10)}

Solve[{1-(x/5)-y==0,-1+2 x-((3 y)/10)==0},{x,y}]

Linearization

field[x,y][[1]]

matrix[x_,y_] = {{D[field[x,y][[1]],x],D[field[x,y][[1]],y]},{D[field[x,y][[2]],x],D[field[x,y][[2]],y]}}

matrix[65/103,90/103]

Det[matrix[65/103,90/103]-lambda {{1,0},{0,1}}]

Solve[Det[matrix[65/103,90/103]-lambda {{1,0},{0,1}}]==0,lambda]

Numerical Solution

The following lines contain what may be the most practically useful information presented in this course.

soln[xzero_,yzero_]:=NDSolve[{odex'[t]==field[odex[t],odey[t]][[1]],odey'[t]==field[odex[t],odey[t]][[2]],odex[0]==xzero,odey[0]==yzero},{odex,odey},{t,0,10}]

soln[1,1]

onesoln[t_]={odex[t]/.soln[1,1][[1]],odey[t]/.soln[1,1][[1]]}

Show[interest,ParametricPlot[onesoln[t],{t,0,10},PlotStyle->{Thick,Red}]]

solnlong[xzero_,yzero_]:=NDSolve[{odex'[t]==field[odex[t],odey[t]][[1]],odey'[t]==field[odex[t],odey[t]][[2]],odex[0]==xzero,odey[0]==yzero},{odex,odey},{t,0,100}]

onesolnlong[t_]={odex[t]/.solnlong[1,1][[1]],odey[t]/.solnlong[1,1][[1]]}

Show[interest,ParametricPlot[onesolnlong[t],{t,0,50},PlotStyle->{Thick,Red}]]

Animate[Show[interest,ParametricPlot[onesolnlong[t],{t,0,end},PlotStyle->{Thick,Red}]],{end,0.1,100}]

Another Equilibrium Point

field[5,0]

matrix[5,0]

matrix[0,-10/3]