[PDF] Solving Systems of Differential Equations PDF worksheet7

In the command window, type: global a and in the system ex m file, change it to function [yprime] = system ex(t, y) global a yprime(1,1) = y(2); yprime(2,1) = a*y(2)-sin(y(1)); In the command window, set a equal to whatever value you'd like, and plot the solutions using ode45

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Solving Systems of Dierential Equations

1 Solving Systems of Dierential Equations

We know how to useode45to solve a rst order dierential equation, but it can handle much more than this. We will now go over how to solve systems of dierential equations using Matlab.

Consider the system of dierential equations

y 01=y2 y 02=15 y2sin(y1) We would like to solve this forward in time. To do this, we must rst create a function M-File that holds the dierential equation. It works exactly how the function M-le works for solving a rst-order dierential equation, except we must treat our variables (except time) as vectors instead of scalars as we did before. The function M-File for this dierential equation should be saved as systemex.mand looks likefunctiony prime= s ystem_ex(t,y) yprime = z eros (2,1); yprime(1) = y(2); yprime(2) = -1/5*y(2)- sin (y(1)); See howyis a vector, wherey(1)is associated withy1andy(2)is associated withy2? The same is true ofyprime, whereyprime(1)is associated withy1andyprime(2)is associated withy2.

That's all there is to it!

Now we'd like to solve the dierential equation with initial conditionsy1(0) = 0 andy2(0) = 3 forward in time, lets sayt2[0;40]. The command is just the same as we have used before, except we need to give it a vector of initial conditions instead of just a scalar. In the command window, type [t,y] = ode45(@systemex,[0,40],[0,3]) The system has been numerically solved. Looking in the workspace, you see we now have two variables.tholds all the time steps whileyis a matrix with 2 columns. The rst column of the matrix is all they1values and the second column is all they2values. You can plot these against time to see the solution of each variable, or plot them against each other to generate solutions in the phase plane: plot(t,y(:,1)) plot(t,y(:,2)) plot(y(:,1),y(:,2))

Try this with some more initial conditions.

2 Global Variables

Sometimes, we would like to have a parameter inside our function m-le. To do this, we declare a global variable, since it's hard to pass these usingode45. Say we now have the system: y 01=y2 y

02=ay2sin(y1)

whereais a parameter. In the command window, type: global a and in thesystemex.mle, change it to function [yprime] = systemex(t, y) global a yprime(1,1) = y(2); yprime(2,1) = a*y(2)-sin(y(1)); In the command window, setaequal to whatever value you'd like, and plot the solutions using ode45. You can see that the value is automatically changed insystemex.mwhenever you change it in the command window. Alternatively, instead of using global variables we could changesystemex.mto: function [yprime] = systemex(t,y,a) yprime(1,1)=y(2); yprime(2,1)=a*y(2)-sin(y(1)); and in the command window type: [t,y] = ode45(@systemex,[0,40],[0,3],[],-1/5)

3 Contour Plots

Matlab can generate contour plots quite easily. First we create a mesh usingmeshgrid. Then we use the contour command to plot the contours of the given equation. If we wanted to plot the contours for the equation of a circlex2+y2for values ofxandyin the unit circle, we type [x,y]=meshgrid(-1:0.01:1,-1:0.01:1); contour(x,y,x.^2+y.^2,20) Type help contour to see all the optional parameters.

4 Homework #10

Solve the system of equations

0(t) =sin(x(t)) +y(t)

0(t) =cos(x(t))15

y(t) usingode45, over the time intervalt2(0:40) with initial conditionsx(0) = 4 andy(0) = 0. Then, plotyagainstx. You should observe the trajectory approaching an equilibrium , i.e. a single point (x0;y0), in this space. (Hint: Generally, we plotyagainsttusingplot(t,y). How would we plot yagainstx?) 2quotesdbs_dbs4.pdfusesText_8

[PDF] [PDF] Solving Systems of Differential Equations