Systems of Differential Equations Math 3740 Summer 2008 Jay Treiman LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= This worksheet is meant start students looking at the behavior of solutions to systems of differential equations. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

restart; with(DEtools): with(LinearAlgebra): with(plots):

One can use the dsolve command solve systems of differential equations. Here is an example of taking a third order differential equation that is transformed into a system of three first order differential equations. The solutions we get are compared to the solutions to the third order differential equation. The differential equation is 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 One then defines new vaiables LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSI9RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSS8lKXN0cmV0Y2h5R0ZJLyUqc3ltbWV0cmljR0ZJLyUobGFyZ2VvcEdGSS8lLm1vdmFibGVsaW1pdHNHRkkvJSdhY2NlbnRHRkkvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZYLUYvNiVRInhGJ0YyRjVGPg==, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYyLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSI9RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSS8lKXN0cmV0Y2h5R0ZJLyUqc3ltbWV0cmljR0ZJLyUobGFyZ2VvcEdGSS8lLm1vdmFibGVsaW1pdHNHRkkvJSdhY2NlbnRHRkkvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZYLUYvNiVRInhGJ0YyRjUtRkQ2LVEiJ0YnRj5GR0ZKRkxGTkZQRlJGVC9GV1EsMC4xMTExMTExZW1GJy9GWlEmMC4wZW1GJy1GRDYtUSIsRidGPkZHL0ZLRjRGTEZORlBGUkZUL0ZXRl5vL0ZaUSwwLjMzMzMzMzNlbUYnLUZENi1RIn5GJ0Y+RkdGSkZMRk5GUEZSRlRGY29GXW8tRkQ2L1EkYW5kRicvJSVib2xkR0Y0L0Y2USVib2xkRicvJStmb250d2VpZ2h0R0ZfcEZHRkpGTEZORlBGUkZURmNvRl1vRmZvRmZvLUYsNiVGLi1GIzYkLUY7NiRRIjNGJ0Y+Rj5GQEZDRmZvRmVuRmhuRmhuRj4=. The system of differential equations now becomes DEQs_1 := {diff(z1(t),t)=z2(t), diff(z2(t),t)=z3(t), diff(z3(t),t)=-8*z1(t)+10*z2(t)-z3(t)}; One can then use dsolve. solns_1 := dsolve(DEQs_1); Since LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg== is the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the differential equation, the solution is subs(solns_1,z1(t)); One can compare this with the solution to the third order differential equation to see that they are the same. dsolve((diff(x(t), t, t, t))+(diff(x(t), t, t))-10*(diff(x(t), t))+8*x(t) = 0);

Solving a system of differential equations in the form 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 where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY5is an LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJ4RidGL0YyRjVGK0Y5 matrix and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is an LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= vector, can be done by Maple. Here is a simple matrix for a differential equation 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 A := Matrix([[1,2],[2,1]]); One can use the matrixDE command to solve the system of differential equations. A := Matrix([[1,2],[2,1]]); solns_2 := matrixDE(A,t); This tells us that there are two solutions. SubMatrix(convert(solns_2[1],Matrix),1..2,1); SubMatrix(convert(solns_2[1],Matrix),1..2,2);

One can also work directly by finding the eigenvalues and eigenvectors for the matrixx LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= to construct the solutions. First we get the characteristic polynomial. M1 := A-lambda*IdentityMatrix(2); CharPoly1 := Determinant(M1); One the soles the charateristic equation. eig_vals := [solve(CharPoly1,lambda)]; One can the get the eigenvectors by solving 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. v_2 := [op(NullSpace(subs(lambda=eig_vals[1],M1))), op(NullSpace(subs(lambda=eig_vals[2],M1)))]; These can then be put together into the solutions. add(cat(c,i)*v_2[i]*exp(eig_vals[i]*t),i=1..2);

tOne can also work directly by finding the eigenvalues and eigenvectors for the matrixx LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= to construct the solutions. We will work with the system of differential equations 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 DEQs_3 :={D(x)(t)=0.01*x(t)-0.001*x(t)*(x(t)+y(t)), D(y)(t)=-0.0005*x(t)*y(t)+0.02*y(t)-0.001*y(t)*z(t), D(z)(t)=-0.0001*z(t)+0.0005*y(t)*z(t)}; One can use DEplot and DEplot3d to visualize some of the solutions. Here are some examples with DEplot3d. DEplot3d(DEQs_3,[x(t),y(t),z(t)],t=0..1000, [[x(0)=1,y(0)=10,z(0)=1],[x(0)=10,y(0)=10,z(0)=20], [x(0)=10,y(0)=1,z(0)=1],[x(0)=20,y(0)=10,z(0)=5], [x(0)=10,y(0)=10,z(0)=10]],linecolor=t,stepsize=0.2); One can look at two of the variables instead of three. DEplot(DEQs_3,[x(t),y(t),z(t)],t=0..1000, [[x(0)=1,y(0)=10,z(0)=1],[x(0)=10,y(0)=10,z(0)=20], [x(0)=10,y(0)=1,z(0)=1],[x(0)=20,y(0)=10,z(0)=5], [x(0)=10,y(0)=10,z(0)=10]],scene=[x(t),z(t)], linecolor=t,stepsize=0.2); One can use the odeplot combined with dsolve to show all of the variables plotted against LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= at the same time for one set of initial conditions. p := dsolve({op(DEQs_3),x(0)=10,y(0)=15,z(0)=5},{x(t),y(t),z(t)}, type=numeric); odeplot(p,[[t,x(t),color=red,linestyle=DASH,thickness=1], [t,y(t),color=khaki,linestyle=DASHDOT,thickness=1], [t,z(t),color=blue, thickness=1]],0..1000);

Find the eigenvalues and eigenvectors for the matrix 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

Use Maple to solve the system of differential equations 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

Plot at least five different solutions for the system of differential equations 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 and 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 What can you say about the what happens to all solutions to the system over time?

What happens to the solutions to the differential syatem of differential equations 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and 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 What can you say about the what happens to all solutions to the system over time? Compare this to the system in Exercise 3, which is the same sysem when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkjbW5HRiQ2JFEiMEYnL0YzUSdub3JtYWxGJ0Y+Rj4tSSNtb0dGJDYtUSI9RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRi8lKXN0cmV0Y2h5R0ZGLyUqc3ltbWV0cmljR0ZGLyUobGFyZ2VvcEdGRi8lLm1vdmFibGVsaW1pdHNHRkYvJSdhY2NlbnRHRkYvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZVLUY7NiRRIzAuRidGPkY+