Solving ordinary differential equations

This file contains functions useful for solving differential equations which occur commonly in a 1st semester differential equations course. For another numerical solver see ode_solver() function and optional package Octave.

Commands:

  • desolve - Computes the “general solution” to a 1st or 2nd order ODE via Maxima.
  • desolve_laplace - Solves an ODE using laplace transforms via Maxima. Initials conditions are optional.
  • desolve_system - Solves any size system of 1st order odes using Maxima. Initials conditions are optional.
  • desolve_rk4 - Solves numerically IVP for one first order equation, returns list of points or plot
  • desolve_system_rk4 - Solves numerically IVP for system of first order equations, returns list of points
  • desolve_odeint - Solves numerically a system of first-order ordinary differential equations using odeint from scipy.integrate module.
  • eulers_method - Approximate solution to a 1st order DE, presented as a table.
  • eulers_method_2x2 - Approximate solution to a 1st order system of DEs, presented as a table.
  • eulers_method_2x2_plot - Plots the sequence of points obtained from Euler’s method.

AUTHORS:

  • David Joyner (3-2006) - Initial version of functions
  • Marshall Hampton (7-2007) - Creation of Python module and testing
  • Robert Bradshaw (10-2008) - Some interface cleanup.
  • Robert Marik (10-2009) - Some bugfixes and enhancements
sage.calculus.desolvers.desolve(de, dvar, ics=None, ivar=None, show_method=False, contrib_ode=False)

Solves a 1st or 2nd order linear ODE via maxima. Including IVP and BVP.

Use desolve? <tab> if the output in truncated in notebook.

INPUT:

  • de - an expression or equation representing the ODE
  • dvar - the dependent variable (hereafter called y)
  • ics - (optional) the initial or boundary conditions
    • for a first-order equation, specify the initial x and y
    • for a second-order equation, specify the initial x, y, and dy/dx, i.e. write \([x_0, y(x_0), y'(x_0)]\)
    • for a second-order boundary solution, specify initial and final x and y boundary conditions, i.e. write \([x_0, y(x_0), x_1, y(x_1)]\).
    • gives an error if the solution is not SymbolicEquation (as happens for example for Clairaut equation)
  • ivar - (optional) the independent variable (hereafter called x), which must be specified if there is more than one independent variable in the equation.
  • show_method - (optional) if true, then Sage returns pair [solution, method], where method is the string describing method which has been used to get solution (Maxima uses the following order for first order equations: linear, separable, exact (including exact with integrating factor), homogeneous, bernoulli, generalized homogeneous) - use carefully in class, see below for the example of the equation which is separable but this property is not recognized by Maxima and equation is solved as exact.
  • contrib_ode - (optional) if true, desolve allows to solve clairaut, lagrange, riccati and some other equations. May take a long time and thus turned off by default. Initial conditions can be used only if the result is one SymbolicEquation (does not contain singular solution, for example)

OUTPUT:

In most cases returns SymbolicEquation which defines the solution implicitly. If the result is in the form y(x)=... (happens for linear eqs.), returns the right-hand side only. The possible constant solutions of separable ODE’s are omitted.

EXAMPLES:

sage: x = var('x')
sage: y = function('y', x)
sage: desolve(diff(y,x) + y - 1, y)
(c + e^x)*e^(-x)
sage: f = desolve(diff(y,x) + y - 1, y, ics=[10,2]); f
(e^10 + e^x)*e^(-x)
sage: plot(f)

We can also solve second-order differential equations.:

sage: x = var('x')
sage: y = function('y', x)
sage: de = diff(y,x,2) - y == x
sage: desolve(de, y)
k2*e^(-x) + k1*e^x - x
sage: f = desolve(de, y, [10,2,1]); f
-x + 7*e^(x - 10) + 5*e^(-x + 10)
sage: f(x=10)
2
sage: diff(f,x)(x=10)
1
sage: de = diff(y,x,2) + y == 0
sage: desolve(de, y)
k2*cos(x) + k1*sin(x)
sage: desolve(de, y, [0,1,pi/2,4])
cos(x) + 4*sin(x)
sage: desolve(y*diff(y,x)+sin(x)==0,y)
-1/2*y(x)^2 == c - cos(x)

Clairot equation: general and singular solutions:

sage: desolve(diff(y,x)^2+x*diff(y,x)-y==0,y,contrib_ode=True,show_method=True)
[[y(x) == c^2 + c*x, y(x) == -1/4*x^2], 'clairault']

For equations involving more variables we specify independent variable:

sage: a,b,c,n=var('a b c n')
sage: desolve(x^2*diff(y,x)==a+b*x^n+c*x^2*y^2,y,ivar=x,contrib_ode=True)
[[y(x) == 0, (b*x^(n - 2) + a/x^2)*c^2*u == 0]]
sage: desolve(x^2*diff(y,x)==a+b*x^n+c*x^2*y^2,y,ivar=x,contrib_ode=True,show_method=True)
[[[y(x) == 0, (b*x^(n - 2) + a/x^2)*c^2*u == 0]], 'riccati']

Higher orded, not involving independent variable:

sage: desolve(diff(y,x,2)+y*(diff(y,x,1))^3==0,y).expand()
1/6*y(x)^3 + k1*y(x) == k2 + x
sage: desolve(diff(y,x,2)+y*(diff(y,x,1))^3==0,y,[0,1,1,3]).expand()
1/6*y(x)^3 - 5/3*y(x) == x - 3/2
sage: desolve(diff(y,x,2)+y*(diff(y,x,1))^3==0,y,[0,1,1,3],show_method=True)
[1/6*y(x)^3 - 5/3*y(x) == x - 3/2, 'freeofx']

Separable equations - Sage returns solution in implicit form:

sage: desolve(diff(y,x)*sin(y) == cos(x),y)
-cos(y(x)) == c + sin(x)
sage: desolve(diff(y,x)*sin(y) == cos(x),y,show_method=True)
[-cos(y(x)) == c + sin(x), 'separable']
sage: desolve(diff(y,x)*sin(y) == cos(x),y,[pi/2,1])
-cos(y(x)) == -cos(1) + sin(x) - 1

Linear equation - Sage returns the expression on the right hand side only:

sage: desolve(diff(y,x)+(y) == cos(x),y)
1/2*((cos(x) + sin(x))*e^x + 2*c)*e^(-x)
sage: desolve(diff(y,x)+(y) == cos(x),y,show_method=True)
[1/2*((cos(x) + sin(x))*e^x + 2*c)*e^(-x), 'linear']
sage: desolve(diff(y,x)+(y) == cos(x),y,[0,1])
1/2*(cos(x)*e^x + e^x*sin(x) + 1)*e^(-x)

This ODE with separated variables is solved as exact. Explanation - factor does not split \(e^{x-y}\) in Maxima into \(e^{x}e^{y}\):

sage: desolve(diff(y,x)==exp(x-y),y,show_method=True)
[-e^x + e^y(x) == c, 'exact']

You can solve Bessel equations. You can also use initial conditions, but you cannot put (sometimes desired) initial condition at x=0, since this point is singlar point of the equation. Anyway, if the solution should be bounded at x=0, then k2=0.:

sage: desolve(x^2*diff(y,x,x)+x*diff(y,x)+(x^2-4)*y==0,y)
k1*bessel_J(2, x) + k2*bessel_Y(2, x)

Difficult ODE produces error:

sage: desolve(sqrt(y)*diff(y,x)+e^(y)+cos(x)-sin(x+y)==0,y) # not tested
Traceback (click to the left for traceback)
...
NotImplementedError, "Maxima was unable to solve this ODE. Consider to set option contrib_ode to True."

Difficult ODE produces error - moreover, takes a long time

sage: desolve(sqrt(y)*diff(y,x)+e^(y)+cos(x)-sin(x+y)==0,y,contrib_ode=True) # not tested

Some more types od ODE’s:

sage: desolve(x*diff(y,x)^2-(1+x*y)*diff(y,x)+y==0,y,contrib_ode=True,show_method=True)
[[y(x) == c + log(x), y(x) == c*e^x], 'factor']
sage: desolve(diff(y,x)==(x+y)^2,y,contrib_ode=True,show_method=True)
[[[x == c - arctan(sqrt(t)), y(x) == -x - sqrt(t)], [x == c + arctan(sqrt(t)), y(x) == -x + sqrt(t)]], 'lagrange']

These two examples produce error (as expected, Maxima 5.18 cannot solve equations from initial conditions). Current Maxima 5.18 returns false answer in this case!:

sage: desolve(diff(y,x,2)+y*(diff(y,x,1))^3==0,y,[0,1,2]).expand() # not tested
Traceback (click to the left for traceback)
...
NotImplementedError, "Maxima was unable to solve this ODE. Consider to set option contrib_ode to True."
sage: desolve(diff(y,x,2)+y*(diff(y,x,1))^3==0,y,[0,1,2],show_method=True) # not tested
Traceback (click to the left for traceback)
...
NotImplementedError, "Maxima was unable to solve this ODE. Consider to set option contrib_ode to True."

Second order linear ODE:

sage: desolve(diff(y,x,2)+2*diff(y,x)+y == cos(x),y)
(k2*x + k1)*e^(-x) + 1/2*sin(x)
sage: desolve(diff(y,x,2)+2*diff(y,x)+y == cos(x),y,show_method=True)
[(k2*x + k1)*e^(-x) + 1/2*sin(x), 'variationofparameters']
sage: desolve(diff(y,x,2)+2*diff(y,x)+y == cos(x),y,[0,3,1])
1/2*(7*x + 6)*e^(-x) + 1/2*sin(x)
sage: desolve(diff(y,x,2)+2*diff(y,x)+y == cos(x),y,[0,3,1],show_method=True)
[1/2*(7*x + 6)*e^(-x) + 1/2*sin(x), 'variationofparameters']
sage: desolve(diff(y,x,2)+2*diff(y,x)+y == cos(x),y,[0,3,pi/2,2])
3*(x*(e^(1/2*pi) - 2)/pi + 1)*e^(-x) + 1/2*sin(x)
sage: desolve(diff(y,x,2)+2*diff(y,x)+y == cos(x),y,[0,3,pi/2,2],show_method=True)
[3*(x*(e^(1/2*pi) - 2)/pi + 1)*e^(-x) + 1/2*sin(x), 'variationofparameters']
sage: desolve(diff(y,x,2)+2*diff(y,x)+y == 0,y)
(k2*x + k1)*e^(-x)
sage: desolve(diff(y,x,2)+2*diff(y,x)+y == 0,y,show_method=True)
[(k2*x + k1)*e^(-x), 'constcoeff']
sage: desolve(diff(y,x,2)+2*diff(y,x)+y == 0,y,[0,3,1])
(4*x + 3)*e^(-x)
sage: desolve(diff(y,x,2)+2*diff(y,x)+y == 0,y,[0,3,1],show_method=True)
[(4*x + 3)*e^(-x), 'constcoeff']
sage: desolve(diff(y,x,2)+2*diff(y,x)+y == 0,y,[0,3,pi/2,2])
(2*x*(2*e^(1/2*pi) - 3)/pi + 3)*e^(-x)
sage: desolve(diff(y,x,2)+2*diff(y,x)+y == 0,y,[0,3,pi/2,2],show_method=True)
[(2*x*(2*e^(1/2*pi) - 3)/pi + 3)*e^(-x), 'constcoeff']

TESTS:

Trac #9961 fixed (allow assumptions on the dependent variable in desolve):

sage: y=function('y',x); assume(x>0); assume(y>0)
sage: sage.calculus.calculus.maxima('domain:real')  # needed since Maxima 5.26.0 to get the answer as below
real
sage: desolve(x*diff(y,x)-x*sqrt(y^2+x^2)-y == 0, y, contrib_ode=True)
[x - arcsinh(y(x)/x) == c]

Trac #10682 updated Maxima to 5.26, and it started to show a different solution in the complex domain for the ODE above:

sage: sage.calculus.calculus.maxima('domain:complex')  # back to the default complex domain
complex
sage: desolve(x*diff(y,x)-x*sqrt(y^2+x^2)-y == 0, y, contrib_ode=True)
[1/2*(2*x^2*sqrt(x^(-2)) - 2*x*sqrt(x^(-2))*arcsinh(y(x)/sqrt(x^2)) -
    2*x*sqrt(x^(-2))*arcsinh(y(x)^2/(x*sqrt(y(x)^2))) +
    log(4*(2*x^2*sqrt((x^2*y(x)^2 + y(x)^4)/x^2)*sqrt(x^(-2)) + x^2 +
    2*y(x)^2)/x^2))/(x*sqrt(x^(-2))) == c]

Trac #6479 fixed:

sage: x = var('x')
sage: y = function('y', x)
sage: desolve( diff(y,x,x) == 0, y, [0,0,1])
x
sage: desolve( diff(y,x,x) == 0, y, [0,1,1])
x + 1

Trac #9835 fixed:

sage: x = var('x')
sage: y = function('y', x)
sage: desolve(diff(y,x,2)+y*(1-y^2)==0,y,[0,-1,1,1])
Traceback (most recent call last):
...
NotImplementedError: Unable to use initial condition for this equation (freeofx).

Trac #8931 fixed:

sage: x=var('x'); f=function('f',x); k=var('k'); assume(k>0)
sage: desolve(diff(f,x,2)/f==k,f,ivar=x)
k1*e^(sqrt(k)*x) + k2*e^(-sqrt(k)*x)

AUTHORS:

  • David Joyner (1-2006)
  • Robert Bradshaw (10-2008)
  • Robert Marik (10-2009)
sage.calculus.desolvers.desolve_laplace(de, dvar, ics=None, ivar=None)

Solves an ODE using laplace transforms. Initials conditions are optional.

INPUT:

  • de - a lambda expression representing the ODE (eg, de = diff(y,x,2) == diff(y,x)+sin(x))
  • dvar - the dependent variable (eg y)
  • ivar - (optional) the independent variable (hereafter called x), which must be specified if there is more than one independent variable in the equation.
  • ics - a list of numbers representing initial conditions, (eg, f(0)=1, f’(0)=2 is ics = [0,1,2])

OUTPUT:

Solution of the ODE as symbolic expression

EXAMPLES:

sage: u=function('u',x)
sage: eq = diff(u,x) - exp(-x) - u == 0
sage: desolve_laplace(eq,u)
1/2*(2*u(0) + 1)*e^x - 1/2*e^(-x)

We can use initial conditions:

sage: desolve_laplace(eq,u,ics=[0,3])
-1/2*e^(-x) + 7/2*e^x

The initial conditions do not persist in the system (as they persisted in previous versions):

sage: desolve_laplace(eq,u)
1/2*(2*u(0) + 1)*e^x - 1/2*e^(-x)
sage: f=function('f', x)
sage: eq = diff(f,x) + f == 0
sage: desolve_laplace(eq,f,[0,1])
e^(-x)
sage: x = var('x')
sage: f = function('f', x)
sage: de = diff(f,x,x) - 2*diff(f,x) + f
sage: desolve_laplace(de,f)
-x*e^x*f(0) + x*e^x*D[0](f)(0) + e^x*f(0)
sage: desolve_laplace(de,f,ics=[0,1,2])
x*e^x + e^x

TESTS:

Trac #4839 fixed:

sage: t=var('t')
sage: x=function('x', t)
sage: soln=desolve_laplace(diff(x,t)+x==1, x, ics=[0,2])
sage: soln
e^(-t) + 1
sage: soln(t=3)
e^(-3) + 1

AUTHORS:

  • David Joyner (1-2006,8-2007)
  • Robert Marik (10-2009)
sage.calculus.desolvers.desolve_odeint(des, ics, times, dvars, ivar=None, compute_jac=False, args=(), rtol=None, atol=None, tcrit=None, h0=0.0, hmax=0.0, hmin=0.0, ixpr=0, mxstep=0, mxhnil=0, mxordn=12, mxords=5, printmessg=0)

Solves numerically a system of first-order ordinary differential equations using odeint from scipy.integrate module.

INPUT:

  • des – right hand sides of the system
  • ics – initial conditions
  • times – a sequence of time points in which the solution must be found
  • dvars – dependent variables. ATTENTION: the order must be the same as in des, that means: d(dvars[i])/dt=des[i]
  • ivar – independent variable, optional.
  • compute_jac – boolean. If True, the Jacobian of des is computed and used during the integration of Stiff Systems. Default value is False.
Other Parameters (taken from the documentation of odeint function from
scipy.integrate module)
  • rtol, atol : float The input parameters rtol and atol determine the error control performed by the solver. The solver will control the vector, e, of estimated local errors in y, according to an inequality of the form:

    max-norm of (e / ewt) <= 1

    where ewt is a vector of positive error weights computed as:

    ewt = rtol * abs(y) + atol

    rtol and atol can be either vectors the same length as y or scalars.

  • tcrit : array Vector of critical points (e.g. singularities) where integration care should be taken.

  • h0 : float, (0: solver-determined) The step size to be attempted on the first step.

  • hmax : float, (0: solver-determined) The maximum absolute step size allowed.

  • hmin : float, (0: solver-determined) The minimum absolute step size allowed.

  • ixpr : boolean. Whether to generate extra printing at method switches.

  • mxstep : integer, (0: solver-determined) Maximum number of (internally defined) steps allowed for each integration point in t.

  • mxhnil : integer, (0: solver-determined) Maximum number of messages printed.

  • mxordn : integer, (0: solver-determined) Maximum order to be allowed for the nonstiff (Adams) method.

  • mxords : integer, (0: solver-determined) Maximum order to be allowed for the stiff (BDF) method.

OUTPUT:

Returns a list with the solution of the system at each time in times.

EXAMPLES:

Lotka Volterra Equations:

sage: from sage.calculus.desolvers import desolve_odeint
sage: x,y=var('x,y')
sage: f=[x*(1-y),-y*(1-x)]
sage: sol=desolve_odeint(f,[0.5,2],srange(0,10,0.1),[x,y])
sage: p=line(zip(sol[:,0],sol[:,1]))
sage: p.show()

Lorenz Equations:

sage: x,y,z=var('x,y,z')
sage: # Next we define the parameters
sage: sigma=10
sage: rho=28
sage: beta=8/3
sage: # The Lorenz equations
sage: lorenz=[sigma*(y-x),x*(rho-z)-y,x*y-beta*z]
sage: # Time and initial conditions
sage: times=srange(0,50.05,0.05)
sage: ics=[0,1,1]
sage: sol=desolve_odeint(lorenz,ics,times,[x,y,z],rtol=1e-13,atol=1e-14)

One-dimensional Stiff system:

sage: y= var('y')
sage: epsilon=0.01
sage: f=y^2*(1-y)
sage: ic=epsilon
sage: t=srange(0,2/epsilon,1)
sage: sol=desolve_odeint(f,ic,t,y,rtol=1e-9,atol=1e-10,compute_jac=True)
sage: p=points(zip(t,sol))
sage: p.show()

Another Stiff system with some optional parameters with no default value:

sage: y1,y2,y3=var('y1,y2,y3')
sage: f1=77.27*(y2+y1*(1-8.375*1e-6*y1-y2))
sage: f2=1/77.27*(y3-(1+y1)*y2)
sage: f3=0.16*(y1-y3)
sage: f=[f1,f2,f3]
sage: ci=[0.2,0.4,0.7]
sage: t=srange(0,10,0.01)
sage: v=[y1,y2,y3]
sage: sol=desolve_odeint(f,ci,t,v,rtol=1e-3,atol=1e-4,h0=0.1,hmax=1,hmin=1e-4,mxstep=1000,mxords=17)

AUTHOR:

  • Oriol Castejon (05-2010)
sage.calculus.desolvers.desolve_rk4(de, dvar, ics=None, ivar=None, end_points=None, step=0.1, output='list', **kwds)

Solves numerically one first-order ordinary differential equation. See also ode_solver.

INPUT:

input is similar to desolve command. The differential equation can be written in a form close to the plot_slope_field or desolve command

  • Variant 1 (function in two variables)
    • de - right hand side, i.e. the function \(f(x,y)\) from ODE \(y'=f(x,y)\)
    • dvar - dependent variable (symbolic variable declared by var)
  • Variant 2 (symbolic equation)
    • de - equation, including term with diff(y,x)
    • dvar` - dependent variable (declared as funciton of independent variable)
  • Other parameters
    • ivar - should be specified, if there are more variables or if the equation is autonomous
    • ics - initial conditions in the form [x0,y0]
    • end_points - the end points of the interval
      • if end_points is a or [a], we integrate on between min(ics[0],a) and max(ics[0],a)
      • if end_points is None, we use end_points=ics[0]+10
      • if end_points is [a,b] we integrate on between min(ics[0],a) and max(ics[0],b)
    • step - (optional, default:0.1) the length of the step (positive number)
    • output - (optional, default: ‘list’) one of ‘list’, ‘plot’, ‘slope_field’ (graph of the solution with slope field)

OUTPUT:

Returns a list of points, or plot produced by list_plot, optionally with slope field.

EXAMPLES:

sage: from sage.calculus.desolvers import desolve_rk4

Variant 2 for input - more common in numerics:

sage: x,y=var('x y')
sage: desolve_rk4(x*y*(2-y),y,ics=[0,1],end_points=1,step=0.5)
[[0, 1], [0.5, 1.12419127425], [1.0, 1.46159016229]]

Variant 1 for input - we can pass ODE in the form used by desolve function In this example we integrate bakwards, since end_points < ics[0]:

sage: y=function('y',x)
sage: desolve_rk4(diff(y,x)+y*(y-1) == x-2,y,ics=[1,1],step=0.5, end_points=0)
[[0.0, 8.90425710896], [0.5, 1.90932794536], [1, 1]]

Here we show how to plot simple pictures. For more advanced aplications use list_plot instead. To see the resulting picture use show(P) in Sage notebook.

sage: x,y=var('x y')
sage: P=desolve_rk4(y*(2-y),y,ics=[0,.1],ivar=x,output='slope_field',end_points=[-4,6],thickness=3)

ALGORITHM:

4th order Runge-Kutta method. Wrapper for command rk in Maxima’s dynamics package. Perhaps could be faster by using fast_float instead.

AUTHORS:

  • Robert Marik (10-2009)
sage.calculus.desolvers.desolve_rk4_determine_bounds(ics, end_points=None)

Used to determine bounds for numerical integration.

  • If end_points is None, the interval for integration is from ics[0] to ics[0]+10
  • If end_points is a or [a], the interval for integration is from min(ics[0],a) to max(ics[0],a)
  • If end_points is [a,b], the interval for integration is from min(ics[0],a) to max(ics[0],b)

EXAMPLES:

sage: from sage.calculus.desolvers import desolve_rk4_determine_bounds
sage: desolve_rk4_determine_bounds([0,2],1)
(0, 1)
sage: desolve_rk4_determine_bounds([0,2])
(0, 10)
sage: desolve_rk4_determine_bounds([0,2],[-2])
(-2, 0)
sage: desolve_rk4_determine_bounds([0,2],[-2,4])
(-2, 4)
sage.calculus.desolvers.desolve_system(des, vars, ics=None, ivar=None)

Solves any size system of 1st order ODE’s. Initials conditions are optional.

Onedimensional systems are passed to desolve_laplace().

INPUT:

  • des - list of ODEs
  • vars - list of dependent variables
  • ics - (optional) list of initial values for ivar and vars
  • ivar - (optional) the independent variable, which must be specified if there is more than one independent variable in the equation.

EXAMPLES:

sage: t = var('t')
sage: x = function('x', t)
sage: y = function('y', t)
sage: de1 = diff(x,t) + y - 1 == 0
sage: de2 = diff(y,t) - x + 1 == 0
sage: desolve_system([de1, de2], [x,y])
[x(t) == (x(0) - 1)*cos(t) - (y(0) - 1)*sin(t) + 1,
 y(t) == (y(0) - 1)*cos(t) + (x(0) - 1)*sin(t) + 1]

Now we give some initial conditions:

sage: sol = desolve_system([de1, de2], [x,y], ics=[0,1,2]); sol
[x(t) == -sin(t) + 1, y(t) == cos(t) + 1]
sage: solnx, solny = sol[0].rhs(), sol[1].rhs()
sage: plot([solnx,solny],(0,1))  # not tested
sage: parametric_plot((solnx,solny),(0,1))  # not tested

TESTS:

Trac #9823 fixed:

sage: t = var('t')
sage: x = function('x', t)
sage: de1 = diff(x,t) + 1 == 0
sage: desolve_system([de1], [x])
-t + x(0)

AUTHORS:

  • Robert Bradshaw (10-2008)
sage.calculus.desolvers.desolve_system_rk4(des, vars, ics=None, ivar=None, end_points=None, step=0.1)

Solves numerically system of first-order ordinary differential equations using the 4th order Runge-Kutta method. Wrapper for Maxima command rk. See also ode_solver.

INPUT:

input is similar to desolve_system and desolve_rk4 commands

  • des - right hand sides of the system
  • vars - dependent variables
  • ivar - (optional) should be specified, if there are more variables or if the equation is autonomous and the independent variable is missing
  • ics - initial conditions in the form [x0,y01,y02,y03,....]
  • end_points - the end points of the interval
    • if end_points is a or [a], we integrate on between min(ics[0],a) and max(ics[0],a)
    • if end_points is None, we use end_points=ics[0]+10
    • if end_points is [a,b] we integrate on between min(ics[0],a) and max(ics[0],b)
  • step – (optional, default: 0.1) the length of the step

OUTPUT:

Returns a list of points.

EXAMPLES:

sage: from sage.calculus.desolvers import desolve_system_rk4

Lotka Volterra system:

sage: from sage.calculus.desolvers import desolve_system_rk4
sage: x,y,t=var('x y t')
sage: P=desolve_system_rk4([x*(1-y),-y*(1-x)],[x,y],ics=[0,0.5,2],ivar=t,end_points=20)
sage: Q=[ [i,j] for i,j,k in P]
sage: LP=list_plot(Q)

sage: Q=[ [j,k] for i,j,k in P]
sage: LP=list_plot(Q)

ALGORITHM:

4th order Runge-Kutta method. Wrapper for command rk in Maxima’s dynamics package. Perhaps could be faster by using fast_float instead.

AUTHOR:

  • Robert Marik (10-2009)
sage.calculus.desolvers.desolve_system_strings(des, vars, ics=None)

Solves any size system of 1st order ODE’s. Initials conditions are optional.

This function is obsolete, use desolve_system.

INPUT:

  • de - a list of strings representing the ODEs in maxima notation (eg, de = “diff(f(x),x,2)=diff(f(x),x)+sin(x)”)
  • vars - a list of strings representing the variables (eg, vars = [“s”,”x”,”y”], where s is the independent variable and x,y the dependent variables)
  • ics - a list of numbers representing initial conditions (eg, x(0)=1, y(0)=2 is ics = [0,1,2])

WARNING:

The given ics sets the initial values of the dependent vars in maxima, so subsequent ODEs involving these variables will have these initial conditions automatically imposed.

EXAMPLES:

sage: from sage.calculus.desolvers import desolve_system_strings
sage: s = var('s')
sage: function('x', s)
x(s)
sage: function('y', s)
y(s)
sage: de1 = lambda z: diff(z[0],s) + z[1] - 1
sage: de2 = lambda z: diff(z[1],s) - z[0] + 1
sage: des = [de1([x(s),y(s)]),de2([x(s),y(s)])]
sage: vars = ["s","x","y"]
sage: desolve_system_strings(des,vars)
["(1-'y(0))*sin(s)+('x(0)-1)*cos(s)+1", "('x(0)-1)*sin(s)+('y(0)-1)*cos(s)+1"]
sage: ics = [0,1,-1]
sage: soln = desolve_system_strings(des,vars,ics); soln
['2*sin(s)+1', '1-2*cos(s)']
sage: solnx, solny = map(SR, soln)
sage: RR(solnx(s=3))
1.28224001611973
sage: P1 = plot([solnx,solny],(0,1))
sage: P2 = parametric_plot((solnx,solny),(0,1))

Now type show(P1), show(P2) to view these.

AUTHORS:

  • David Joyner (3-2006, 8-2007)
sage.calculus.desolvers.eulers_method(f, x0, y0, h, x1, algorithm='table')

This implements Euler’s method for finding numerically the solution of the 1st order ODE y' = f(x,y), y(a)=c. The “x” column of the table increments from x0 to x1 by h (so (x1-x0)/h must be an integer). In the “y” column, the new y-value equals the old y-value plus the corresponding entry in the last column.

For pedagogical purposes only.

EXAMPLES:

sage: from sage.calculus.desolvers import eulers_method
sage: x,y = PolynomialRing(QQ,2,"xy").gens()
sage: eulers_method(5*x+y-5,0,1,1/2,1)
     x                    y                  h*f(x,y)
     0                    1                   -2
   1/2                   -1                 -7/4
     1                -11/4                -11/8
sage: x,y = PolynomialRing(QQ,2,"xy").gens()
sage: eulers_method(5*x+y-5,0,1,1/2,1,algorithm="none")
[[0, 1], [1/2, -1], [1, -11/4], [3/2, -33/8]]
sage: RR = RealField(sci_not=0, prec=4, rnd='RNDU')
sage: x,y = PolynomialRing(RR,2,"xy").gens()
sage: eulers_method(5*x+y-5,0,1,1/2,1,algorithm="None")
[[0, 1], [1/2, -1.0], [1, -2.7], [3/2, -4.0]]
sage: RR = RealField(sci_not=0, prec=4, rnd='RNDU')
sage: x,y=PolynomialRing(RR,2,"xy").gens()
sage: eulers_method(5*x+y-5,0,1,1/2,1)
     x                    y                  h*f(x,y)
     0                    1                 -2.0
   1/2                 -1.0                 -1.7
     1                 -2.7                 -1.3
sage: x,y=PolynomialRing(QQ,2,"xy").gens()
sage: eulers_method(5*x+y-5,1,1,1/3,2)
         x                    y                  h*f(x,y)
         1                    1                  1/3
       4/3                  4/3                    1
       5/3                  7/3                 17/9
         2                 38/9                83/27
sage: eulers_method(5*x+y-5,0,1,1/2,1,algorithm="none")
[[0, 1], [1/2, -1], [1, -11/4], [3/2, -33/8]]
sage: pts = eulers_method(5*x+y-5,0,1,1/2,1,algorithm="none")
sage: P1 = list_plot(pts)
sage: P2 = line(pts)
sage: (P1+P2).show()

AUTHORS:

  • David Joyner
sage.calculus.desolvers.eulers_method_2x2(f, g, t0, x0, y0, h, t1, algorithm='table')

This implements Euler’s method for finding numerically the solution of the 1st order system of two ODEs

x' = f(t, x, y), x(t0)=x0.

y' = g(t, x, y), y(t0)=y0.

The “t” column of the table increments from \(t_0\) to \(t_1\) by \(h\) (so \(\\frac{t_1-t_0}{h}\) must be an integer). In the “x” column, the new x-value equals the old x-value plus the corresponding entry in the next (third) column. In the “y” column, the new y-value equals the old y-value plus the corresponding entry in the next (last) column.

For pedagogical purposes only.

EXAMPLES:

sage: from sage.calculus.desolvers import eulers_method_2x2
sage: t, x, y = PolynomialRing(QQ,3,"txy").gens()
sage: f = x+y+t; g = x-y
sage: eulers_method_2x2(f,g, 0, 0, 0, 1/3, 1,algorithm="none")
[[0, 0, 0], [1/3, 0, 0], [2/3, 1/9, 0], [1, 10/27, 1/27], [4/3, 68/81, 4/27]]
sage: eulers_method_2x2(f,g, 0, 0, 0, 1/3, 1)
     t                    x                h*f(t,x,y)                    y           h*g(t,x,y)
     0                    0                         0                    0                    0
   1/3                    0                       1/9                    0                    0
   2/3                  1/9                      7/27                    0                 1/27
     1                10/27                     38/81                 1/27                  1/9
sage: RR = RealField(sci_not=0, prec=4, rnd='RNDU')
sage: t,x,y=PolynomialRing(RR,3,"txy").gens()
sage: f = x+y+t; g = x-y
sage: eulers_method_2x2(f,g, 0, 0, 0, 1/3, 1)
     t                    x                h*f(t,x,y)                    y           h*g(t,x,y)
     0                    0                      0.00                    0                 0.00
   1/3                 0.00                      0.13                 0.00                 0.00
   2/3                 0.13                      0.29                 0.00                0.043
     1                 0.41                      0.57                0.043                 0.15

To numerically approximate \(y(1)\), where \((1+t^2)y''+y'-y=0\), \(y(0)=1\), \(y'(0)=-1\), using 4 steps of Euler’s method, first convert to a system: \(y_1' = y_2\), \(y_1(0)=1\); \(y_2' = \\frac{y_1-y_2}{1+t^2}\), \(y_2(0)=-1\).:

sage: RR = RealField(sci_not=0, prec=4, rnd='RNDU')
sage: t, x, y=PolynomialRing(RR,3,"txy").gens()
sage: f = y; g = (x-y)/(1+t^2)
sage: eulers_method_2x2(f,g, 0, 1, -1, 1/4, 1)
    t                    x                h*f(t,x,y)                    y           h*g(t,x,y)
    0                    1                     -0.25                   -1                 0.50
  1/4                 0.75                     -0.12                -0.50                 0.29
  1/2                 0.63                    -0.054                -0.21                 0.19
  3/4                 0.63                   -0.0078               -0.031                 0.11
    1                 0.63                     0.020                0.079                0.071

To numerically approximate y(1), where \(y''+ty'+y=0\), \(y(0)=1\), \(y'(0)=0\):

sage: t,x,y=PolynomialRing(RR,3,"txy").gens()
sage: f = y; g = -x-y*t
sage: eulers_method_2x2(f,g, 0, 1, 0, 1/4, 1)
     t                    x                h*f(t,x,y)                    y           h*g(t,x,y)
     0                    1                      0.00                    0                -0.25
   1/4                  1.0                    -0.062                -0.25                -0.23
   1/2                 0.94                     -0.11                -0.46                -0.17
   3/4                 0.88                     -0.15                -0.62                -0.10
     1                 0.75                     -0.17                -0.68               -0.015

AUTHORS:

  • David Joyner
sage.calculus.desolvers.eulers_method_2x2_plot(f, g, t0, x0, y0, h, t1)

Plots solution of ODE

This plots the soln in the rectangle (xrange[0],xrange[1]) x (yrange[0],yrange[1]) and plots using Euler’s method the numerical solution of the 1st order ODEs \(x' = f(t,x,y)\), \(x(a)=x_0\), \(y' = g(t,x,y)\), \(y(a) = y_0\).

For pedagogical purposes only.

EXAMPLES:

sage: from sage.calculus.desolvers import eulers_method_2x2_plot

The following example plots the solution to \(\theta''+\sin(\theta)=0\), \(\theta(0)=\frac 34\), \(\theta'(0) = 0\). Type P[0].show() to plot the solution, (P[0]+P[1]).show() to plot \((t,\theta(t))\) and \((t,\theta'(t))\):

sage: f = lambda z : z[2]; g = lambda z : -sin(z[1])
sage: P = eulers_method_2x2_plot(f,g, 0.0, 0.75, 0.0, 0.1, 1.0)

Previous topic

Further examples from Wester’s paper

Next topic

Discrete Wavelet Transform

This Page