This file contains functions useful for solving differential equations which occur commonly in a 1st semester differential equations course. For another numerical solver see the ode_solver() function and the optional package Octave.
Solutions from the Maxima package can contain the three constants _C, _K1, and _K2 where the underscore is used to distinguish them from symbolic variables that the user might have used. You can substitute values for them, and make them into accessible usable symbolic variables, for example with var("_C").
Commands:
AUTHORS:
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:
OUTPUT:
In most cases return a SymbolicEquation which defines the solution implicitly. If the result is in the form y(x)=... (happens for linear eqs.), return 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)
Graphics object consisting of 1 graphics primitive
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)
Clairaut 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 an 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 order equations, 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 in Maxima
into
:
sage: desolve(diff(y,x)==exp(x-y),y,show_method=True)
[-e^x + e^y(x) == _C, 'exact']
You can solve Bessel equations, also using initial conditions, but you cannot put (sometimes desired) the initial condition at x=0, since this point is a singular 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)
Example of difficult ODE producing an 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."
Another difficult ODE with error - moreover, it 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 of 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*e^x, y(x) == _C + log(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 an error (as expected, Maxima 5.18 cannot solve equations from initial conditions). 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 ticket #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 ticket #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 ticket #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 ticket #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 ticket #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)
trac ticket #15775 fixed:
sage: forget()
sage: y = function('y')(x)
sage: desolve(diff(y, x) == sqrt(abs(y)), dvar=y, ivar=x)
sqrt(-y(x))*(sgn(y(x)) - 1) + (sgn(y(x)) + 1)*sqrt(y(x)) == _C + x
AUTHORS:
Solve an ODE using Laplace transforms. Initial conditions are optional.
INPUT:
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:
Solve numerically a system of first order differential equations using the taylor series integrator implemented in mintides.
INPUT:
OUTPUT:
EXAMPLES:
We integrate a periodic orbit of the Kepler problem along 50 periods:
sage: var('t,x,y,X,Y')
(t, x, y, X, Y)
sage: f(t,x,y,X,Y)=[X, Y, -x/(x^2+y^2)^(3/2), -y/(x^2+y^2)^(3/2)]
sage: ics = [0.8, 0, 0, 1.22474487139159]
sage: t = 100*pi
sage: sol = desolve_mintides(f, ics, 0, t, t, 1e-12, 1e-12) # optional -tides
sage: sol # optional -tides # abs tol 1e-5
[[0.000000000000000,
0.800000000000000,
0.000000000000000,
0.000000000000000,
1.22474487139159],
[314.159265358979,
0.800000000028622,
-5.91973525754241e-9,
7.56887091890590e-9,
1.22474487136329]]
ALGORITHM:
Uses TIDES.
REFERENCES:
Solve numerically a system of first-order ordinary differential equations using odeint from scipy.integrate module.
INPUT:
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:
Return 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:
Solve 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
OUTPUT:
Return 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, 1.12419127424558], [1.0, 1.461590162288825]]
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, 8.904257108962112], [0.5, 1.909327945361535], [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:
Used to determine bounds for numerical integration.
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)
Solve any size system of 1st order ODE’s. Initial conditions are optional.
Onedimensional systems are passed to desolve_laplace().
INPUT:
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:
Check that trac ticket #9823 is 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)
Check that trac ticket #16568 is fixed:
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: des = [de1,de2]
sage: ics = [0,1,-1]
sage: vars = [x,y]
sage: sol = desolve_system(des, vars, ics); sol
[x(t) == 2*sin(t) + 1, y(t) == -2*cos(t) + 1]
sage: solx, soly = sol[0].rhs(), sol[1].rhs()
sage: RR(solx(t=3))
1.28224001611973
sage: P1 = plot([solx,soly], (0,1))
sage: P2 = parametric_plot((solx,soly), (0,1))
Now type show(P1), show(P2) to view these plots.
Check that trac ticket #9824 is fixed:
sage: t = var('t')
sage: epsilon = var('epsilon')
sage: x1 = function('x1', t)
sage: x2 = function('x2', t)
sage: de1 = diff(x1,t) == epsilon
sage: de2 = diff(x2,t) == -2
sage: desolve_system([de1, de2], [x1, x2], ivar=t)
[x1(t) == epsilon*t + x1(0), x2(t) == -2*t + x2(0)]
sage: desolve_system([de1, de2], [x1, x2], ics=[1,1], ivar=t)
Traceback (most recent call last):
...
ValueError: Initial conditions aren't complete: number of vars is different from number of dependent variables. Got ics = [1, 1], vars = [x1(t), x2(t)]
AUTHORS:
Solve numerically a 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
OUTPUT:
Return 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:
Solve numerically a system of first order differential equations using the taylor series integrator in arbitrary precission implemented in tides.
INPUT:
OUTPUT:
EXAMPLES:
We integrate the Lorenz equations with Salztman values for the parameters along 10 periodic orbits with 100 digits of precission:
sage: var('t,x,y,z')
(t, x, y, z)
sage: s = 10
sage: r = 28
sage: b = 8/3
sage: f(t,x,y,z)= [s*(y-x),x*(r-z)-y,x*y-b*z]
sage: x0 = -13.7636106821342005250144010543616538641008648540923684535378642921202827747268115852940239346395038284
sage: y0 = -19.5787519424517955388380414460095588661142400534276438649791334295426354746147526415973165506704676171
sage: z0 = 27
sage: T = 15.586522107161747275678702092126960705284805489972439358895215783190198756258880854355851082660142374
sage: sol = desolve_tides_mpfr(f, [x0, y0, z0],0 , T, T, 1e-100, 1e-100, 100) # optional - tides
sage: sol # optional -tides # abs tol 1e-50
[[0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000,
-13.7636106821342005250144010543616538641008648540923684535378642921202827747268115852940239346395038,
-19.5787519424517955388380414460095588661142400534276438649791334295426354746147526415973165506704676,
27.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000],
[15.5865221071617472756787020921269607052848054899724393588952157831901987562588808543558510826601424,
-13.7636106821342005250144010543616538641008648540923684535378642921202827747268115852940239346315658,
-19.5787519424517955388380414460095588661142400534276438649791334295426354746147526415973165506778440,
26.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999636628]]
ALGORITHM:
Uses TIDES.
Warning
This requires the package tides.
REFERENCES:
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:
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 to
by
(so
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 , where
,
,
, using 4 steps of Euler’s method, first
convert to a system:
,
;
,
.:
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 ,
,
:
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:
Plot 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 ,
,
,
.
For pedagogical purposes only.
EXAMPLES:
sage: from sage.calculus.desolvers import eulers_method_2x2_plot
The following example plots the solution to
,
,
. Type P[0].show() to plot the solution,
(P[0]+P[1]).show() to plot
and
:
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)