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Constructions in Sage: A Sage cookbook |  |
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Constructions in Sage: A Sage cookbook | |
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Symbolically solving ODEs can be done using Sage's interface with Maxima. Numerical solution of ODEs can be done using Sage's interface with Octave (an experimental package), or routines in the GSL (Gnu Scientific Library).
You can solve ODE's symbolically in Sage using the Maxima interface
(do not type the ...):
sage: maxima.de_solve('diff(y,x,2) + 3*x = y', ['x','y'], [1,1,1])
y=3*x-2*%e^(x-1)
sage: maxima.de_solve('diff(y,x,2) + 3*x = y', ['x','y'])
y=%k1*%e^x+%k2*%e^-x+3*x
sage: maxima.de_solve('diff(y,x) + 3*x = y', ['x','y'])
y=(%c-3*(-x-1)*%e^-x)*%e^x
sage: maxima.de_solve('diff(y,x) + 3*x = y', ['x','y'],[1,1])
y=-%e^-1*(5*%e^x-3*%e*x-3*%e)
sage: maxima.clear('x'); maxima.clear('f')
sage: maxima.de_solve_laplace("diff(f(x),x,2) = 2*diff(f(x),x)-f(x)",\
... ["x","f"], [0,1,2])
f(x)=x*%e^x+%e^x
sage: maxima.clear('x'); maxima.clear('f')
sage: f = maxima.de_solve_laplace("diff(f(x),x,2) = 2*diff(f(x),x)-f(x)",\
... ["x","f"])
sage: f
f(x)=x*%e^x*('at('diff(f(x),x,1),x=0))-f(0)*x*%e^x+f(0)*%e^x
sage: f.display2d()
!
x d ! x x
f(x) = x %e (-- (f(x))! ) - f(0) x %e + f(0) %e
dx !
!x = 0
If you have Octave and gnuplot
installed,
sage: octave.de_system_plot(['x+y','x-y'], [1,-1], [0,2]) # optional octave required
on the same graph
(the 
-axis is the horizonal axis) of the system of ODEs
for
Another way this system can be solved is to use the command
desolve_system in calculus/examples.
sage: attach os.environ['SAGE_ROOT'] + '/examples/calculus/desolvers.sage' sage: des = ["'diff(x(t),t)=x(t)+y(t)","'diff(y(t),t)=x(t)-y(t)"] sage: vars = ["t","x","y"] sage: ics = [0,1,-1] sage: desolve_system(des,vars,ics) [x(t)=cosh(2^(1/2)*t),y(t)=2*sinh(2^(1/2)*t)/sqrt(2)-cosh(2^(1/2)*t)]
Finally, Sage can solve linear DEs using power series:
sage: R.<t> = PowerSeriesRing(QQ, default_prec=10) sage: a = 2 - 3*t + 4*t^2 + O(t^10) sage: b = 3 - 4*t^2 + O(t^7) sage: f = a.solve_linear_de(prec=5, b=b, f0=3/5) sage: f 3/5 + 21/5*t + 33/10*t^2 - 38/15*t^3 + 11/24*t^4 + O(t^5) sage: f.derivative() - a*f - b O(t^4)
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Constructions in Sage: A Sage cookbook | |
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