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Drawing direction fields and solutions of ODEs with the package "drawdf"

Diagramme mit Maxima

Plotting with Maxima

Maxima

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Contents

These examples and the explanation texts are taken from the Maxima handbook, section "drawdf"

http://maxima.sourceforge.net/docs/manual/en/maxima.html#SEC_Top

Single ODE with one dependent variable

y x diff opts_and_obj drawdf

standard variables x and y

v u diff u v 21 sys opts_and_obj drawdf

custom variables

v u diff u u.min u.max 31 sys v v.min v.max 31 sys opts_and_obj drawdf

custom variables with range

default range: -10..10

System of two ODEs with two dependent variables

x t diff y t diff 21 sys opts_and_obj drawdf

standard variables x, y and t

x t diff y t diff 21 sys u v 21 sys opts_and_obj drawdf

custom variables

u t diff v t diff 21 sys u u.min u.max 31 sys v v.min v.max 31 sys opts_and_obj drawdf

custom variables with range

Additional graphic objects

x y soln_at

p x y 21 sys :

solution lines through a given point

p1 # # solns_at

x y saddle_at

Solution lines

p1 # # saddles_at

x y point_at

p1 # # points_at

Additional graphic options

Options for solution and saddle lines

soln_arrows false ≡

if true, adds small arrows to the solution lines

tstep 0.1 ≡

Integration control parameters

nsteps false ≡

duration 10 ≡

direction both ≡

forward/right, reverse/left, both

field_degree 1 ≡

1: linear, 2: quadratic, solns: compute short solution lines

field_arrows

field_grid 292921 sys ≡

number of arrows in each direction

field_color black ≡

show_field true ≡

field_tstep

field_nsteps

field_duration

Initializations:

restart the Maxima session

load the "drawdf" package

Make sure that the current directory is set to where the current document is

Set default plot options that shall be applied to all plots in this page.

restart MaximaControl Maxima started successfully.

msg drawdf load Maxima :

dir DocumentDirectory CurrentDirectory :

dopt user_preamble set terminal svg noenhanced size 500,500 dynamic ≡ terminal svg ≡ file_name name ≡ background_color #fefefe ≡ 41 sys :

Example 1: no non-default options

name df1 :

x - exp y + dopt drawdf Maxima 0

df1.svg

Example 2: x and y range specified

name df2 :

y 9 x sin * - y 5 / - 21 sys x 1531 sys y 2 - 231 sys dopt drawdf Maxima 0

df2.svg

Legacy type specification of solution line

name df3 :

2 t cos * 1 - y + t y 21 sys t 5 - 1031 sys y 4 - 931 sys trajectory_at 0031 sys dopt drawdf Maxima 0

df3.svg

Specification of solution lines

soln_at and solns_at draw solution curves passing through the specified points, using a slightly enhanced 4th-order Runge Kutta numerical integrator

name df4 :

opt 0 0.1 21 sys 0 0.1 - 21 sys solns_at color blue ≡ 00 soln_at 31 sys :

2 t cos * 1 - y + t y 21 sys t 5 - 1031 sys y 4 - 931 sys dopt opt drawdf Maxima 0

df4.svg

This example does not work in wxmaxima 5.31.1-1 as well

This seems to hang or take too long...

name df5 :

opt field_degree 2 ≡ field_grid 101021 sys ≡ 0 0.1 21 sys 0 0.1 - 21 sys solns_at color blue ≡ 00 soln_at 51 sys :

2 t cos * 1 - y + t y 21 sys t 5 - 1031 sys y 4 - 931 sys dopt opt drawdf Maxima 0

Plotting solution lines with (rather small) arrows

soln_arrows=true adds arrows to the solution curves, and (by default) removes them from the direction field. It also changes the default colors to emphasize the solution curves.

name df6 :

opt soln_arrows true ≡ line_width 4 ≡ 0 0.1 21 sys 0 0.1 - 21 sys 0021 sys solns_at 31 sys :

2 t cos * 1 - y + t y 21 sys t 5 - 1031 sys y 4 - 931 sys dopt opt drawdf Maxima 0

df6.svg

Plotting with increased line width to see the arrows

name df7 :

opt soln_arrows true ≡ line_width 4 ≡ 0.5 0.5 point_at 0.1 0.2 21 sys 0.2 0.1 21 sys 1 0.8 21 sys 0.1 0.1 21 sys 0.6 0.05 21 sys 0.05 0.4 21 sys 1 0.01 21 sys 0.01 0.75 21 sys solns_at 41 sys :

x 1 x - y - (* y 34 / y - x 2 / - (* 21 sys x 0 1.1 31 sys y 0131 sys dopt opt drawdf Maxima 0

df7.svg

The standard duration for time integration is 10. Not all solution curves reach the asymptotic point within that time.

Direction field with short solution lines as arrows

Here, the integration time for solution lines is set to 40

name df8 :

opt field_degree solns ≡ duration 40 ≡ soln_arrows true ≡ line_width 4 ≡ 0.5 0.5 point_at 0.1 0.2 21 sys 0.2 0.1 21 sys 1 0.8 21 sys 0.1 0.1 21 sys 0.6 0.05 21 sys 0.05 0.4 21 sys 1 0.01 21 sys 0.01 0.75 21 sys solns_at 61 sys :

x 1 x - y - (* y 34 / y - x 2 / - (* 21 sys x 0 1.1 31 sys y 0131 sys dopt opt drawdf Maxima 0

df8.svg

Plotting saddle points and separatrices

name df9 :

opt field_degree solns ≡ tstep 0.05 ≡ soln_arrows true ≡ point_size 0.5 ≡ 0021 sys 2 π * 021 sys 2 π * - 021 sys points_at π 021 sys π - 021 sys saddles_at 61 sys :

y 9 x sin * - y 5 / - 21 sys dopt opt drawdf Maxima 0

df9.svg

The same, but with direction field arrows switched off

name df10 :

opt show_field false ≡ tstep 0.05 ≡ soln_arrows true ≡ point_size 0.5 ≡ 0021 sys 2 π * 021 sys 2 π * - 021 sys points_at π 021 sys π - 021 sys saddles_at 61 sys :

y 9 x sin * - y 5 / - 21 sys dopt opt drawdf Maxima 0

df10.svg

Pimping a plot with standard draw options

You could group the option into further nested lists for better readibility

name df11 :

opt field_color gray ≡ key soln 1 ≡ color black ≡ 00 soln_at key soln 2 ≡ color red ≡ 01 soln_at key isocline ≡ color green ≡ line_width 2 ≡ nticks 100 ≡ t cos t sin t 02 π * parametric 121 sys :

x 2^y 2^+ x 2 - 231 sys y 2 - 231 sys dopt opt drawdf Maxima 0

df11.svg