pdeplot pde uxy ini s 22 7 uxy 1 2 y 2 2

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Unformatted text preview: is the dependent variable • range is the range of the independent variable • ini-conds is a list of initial conditions The following is a plot of the function satisfying both the differential equation ode1 and the initial conditions ic1 above. > DEplot( ode1, y(t), 0..20, [ [ ic1 ] ] ); 1 0.8 y(t) 0.6 0.4 0.2 0 5 10 t 15 20 You can refine the plot by specifying a smaller stepsize. > DEplot( ode1, y(t), 0..20, [ [ ic1 ] ], stepsize=0.2 ); 1 0.8 y(t) 0.6 0.4 0.2 0 5 10 t 15 20 If you specify more than one list of initial conditions, DEplot plots a solution for each. > ic2 := y(0)=0, D(y)(0)=1; 262 • Chapter 7: Solving Calculus Problems ic2 := y(0) = 0, D(y )(0) = 1 > DEplot( ode1, y(t), 0..20, [ [ic1], [ic2] ], stepsize=0.2 ); 1.4 1.2 1 y(t) 0.8 0.6 0.4 0.2 5 10 t 15 20 DEplot can also plot solutions to a set of differential equations. > eq1 := diff(y(t),t) + y(t) + x(t) = 0; eq1 := ( d y(t)) + y(t) + x(t) = 0 dt > eq2 := y(t) = diff(x(t), t); eq2 := y(t) = > ini1 := x(0)=0, y(0)=5; d x(t) dt ini1 := x(0) = 0, y(0) = 5 > ini2 := x(0)=0, y(0)=-5; ini2 := x(0) = 0, y(0) = −5 The system {eq1, eq2} has two dependent variables, x(t) and y(t), so you must include a list of dependent variables. 7.2 Ordinary Differential Equations > DEplot( {eq1, eq2}, [x(t), y(t)], -5..5, > [ [ini1], [ini2] ] ); 60 40 y 20 –60 –40 –20 0 –20 –40 –60 20 40 x 60 • 263 Note: DEplot also generates a direction field, as above, whenever it is meaningful to do so. For more details on how to plot ODEs, refer to the ?DEtools,DEplot help page. DEplot3d is the three-dimensional version of DEplot. The basic syntax of DEplot3d is similar to that of DEplot. For details, refer to the ?DEtools,DEplot3d help page. The following is a three-dimensional plot of the system plotted in two dimensions above. > DEplot3d( {eq1, eq2}, [x(t), y(t)], -5..5, > [ [ini1], [ini2] ] ); 40 20 y(t) 0 –20 –40 –60 –40 –20 0 20 x(t) 0 –2 t –4 40 60 4 2 The following is an example of a plot of a system of three differential equations. > eq1 := diff(x(t),t) = y(t)+z(t); eq1 := d x(t) = y(t) + z(t) dt 264 • Chapter 7: Solving Calculus Problems > eq2 := diff(y(t),t) = -x(t)-y(t); eq2 := d y(t) = −y(t) − x(t) dt > eq3 := diff(z(t),t) = x(t)+y(t)-z(t); eq3 := d z(t) = x(t) + y(t) − z(t) dt These are two lists of initial conditions. > ini1 := [x(0)=1, y(0)=0, z(0)=2]; ini1 := [x(0) = 1, y(0) = 0, z(0) = 2] > ini2 := [x(0)=0, y(0)=2, z(0)=-1]; ini2 := [x(0) = 0, y(0) = 2, z(0) = −1] The DEplot3d command plots two solutions to the system of differential equations {eq1, eq2, eq3}, one solution for each list of initial values. > DEplot3d( {eq1, eq2, eq3}, [x(t), y(t), z(t)], t=0..10, > [ini1, ini2], stepsize=0.1, orientation=[-171, 58] ); 2 z 2 x –1 2 –1 y –1 Discontinuous Forcing Functions In many practical instances the forcing function to a system is discontinuous. Maple provides many ways to describe a system in terms of ODEs and include descriptions of discontinuous forcing functions. 7.2 Ordina...
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