CoCalc Public Filesedo marcelo.sagews
Author: Leon Denis
Views : 45
Compute Environment: Ubuntu 18.04 (Deprecated)
sage: y = function('y')(x)
sage: fig1 = plot(desolve(diff(y,x)+2*y==9,y,[0,-2]),x,[0,1.5],
color='black')
sage: fig2 = plot(desolve(diff(y,x)+2*y==9,y,[0,-1]),x,[0,1.5],
color='blue')
sage: fig3 = plot(desolve(diff(y,x)+2*y==9,y,[0,0]),x,[0,1.5],
color='red')
sage: fig4 = plot(desolve(diff(y,x)+2*y==9,y,[0,1]),x,[0,1.5],
color='green')
sage: f(x,y) = (1, 9 - 2*y)
sage: fig5 = plot_vector_field(f,(x,0,1.5),(y,-2,4),figsize=[6,6],
ticks=[[0,0.5,1,1.5],[-1,-2,0,1]],
color='gray')
sage: show(fig1+fig2+fig3+fig4+fig5,frame=False,aspect_ratio=0.15)

sage: y = function('y')(x)
sage: fig1 = plot(desolve(diff(y,x)+2*y==9,y,[0,-2]),x,[0,1.5],
color='gray')
sage: fig2 = plot(desolve(diff(y,x)+2*y==9,y,[0,-1]),x,[0,1.5],
color='blue')
sage: fig3 = plot(desolve(diff(y,x)+2*y==9,y,[0,0]),x,[0,1.5],
color='red')
sage: fig4 = plot(desolve(diff(y,x)+2*y==9,y,[0,1]),x,[0,1.5],
color='green')
sage: f(x,y) = (1, 9 - 2*y)
sage: fig5 = plot_vector_field(f,(x,0, 0.5,1.5),(y,-2,4),figsize=[6,6],
ticks=[[0,0.5,1,1.5],[-1,-2,0,1]])
sage: show(fig1 + fig2 + fig3 + fig4 + fig5, frame = False, aspect_ratio=0.15)

sage: y = function('y')(x)
sage: fig1 = plot(desolve(diff(y,x)+2*y==9,y,[0,-2]),x,[0,1.5],
color='gray', thickness=1)
sage: fig2 = plot(desolve(diff(y,x)+2*y==9,y,[0,-1]),x,[0,1.5],
color='blue',thickness=2)
sage: fig3 = plot(desolve(diff(y,x)+2*y==9,y,[0,0]),x,[0,1.5],
color='red',thickness=3)
sage: fig4 = plot(desolve(diff(y,x)+2*y==9,y,[0,1]),x,[0,1.5],
color='green', thickness=4)
sage: f(x,y) = (1, 9 - 2*y)
sage: fig5 = plot_vector_field(f,(x,0,1.5),(y,-2,4),figsize=[6,6],
ticks=[[-1,0,1,2],[-1,-2,0,1,2]])
sage: show(fig1 + fig2 + fig3 + fig4 + fig5, frame = False )

sage: y = function('y')(x)
sage: fig1 = plot(desolve(diff(y,x)+2*y==9,y,[0,6]),x,[0,1.5],
color='green')
sage: fig2 = plot(desolve(diff(y,x)+2*y==9,y,[0,4.5]),x,[0,1.5],
color='blue')
sage: fig3 = plot(desolve(diff(y,x)+2*y==9,y,[0,0]),x,[0,1.5],
color='red')
sage: f(x,y) = (1, 9 - 2*y)
sage: z,w=var('z,w')
sage: fig4=plot_slope_field( 9-2*w, (z,0.1,1.5), (w,0,6),figsize=[6,6],
ticks=[[0,1,2],[0.0,4.5,6]], color='gray')
sage: show(fig1 + fig2 + fig3 + fig4, frame = False , aspect_ratio=0.17)

sage: x, y = polygens(QQ,'x, y')
sage: fig1 = list_plot(eulers_method(y, 0, 1, 1/3, 5, algorithm = 'none'), marker = 's')
sage: fig2 = list_plot(eulers_method(y, 0, 1, 1/10, 5, algorithm = 'none'),  marker = 'x')
sage: fig3 = list_plot(eulers_method(y, 0, 1, 1/25, 5, algorithm = 'none'), marker = 'o')
sage: t = var('t'); z = function('z')(t)
sage: fig4 = plot(desolve(diff(z, t) == z, z, [0, 1]), t, [0.5, 5])
sage: show(fig1 + fig2 + fig3 + fig4, figsize=[6,3])

scatter_plot

sage: var('s,t')
sage: y=function('y')(t)
sage: eq=diff(y,t,2)+4*diff(y,t)+5*y==(exp(-3*t))*cos(t)
sage: eq.laplace(t,s)

(s, t) s^2*laplace(y(t), t, s) + 4*s*laplace(y(t), t, s) - s*y(0) + 5*laplace(y(t), t, s) - 4*y(0) - D[0](y)(0) == (s + 3)/(s^2 + 6*s + 10)
_

s^2*laplace(y(t), t, s) + 4*s*laplace(y(t), t, s) - s*y(0) + 5*laplace(y(t), t, s) - 4*y(0) - D[0](y)(0) == (s + 3)/(s^2 + 6*s + 10)
_.subs(y(0)=2, D[0](y)(0)=1)

Error in lines 1-1 Traceback (most recent call last): File "/projects/sage/sage-7.5/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 995, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "<string>", line 1 SyntaxError: keyword can't be an expression
eq2 = s^2*laplace(y(t), t, s) + 4*s*laplace(y(t), t, s) - s*y(0) + 5*laplace(y(t), t, s) - 4*y(0) - D[0](y)(0) == (s + 3)/(s^2 + 6*s + 10)

sage: eq2=s^2*laplace(y(t), t, s) + 4*s*laplace(y(t), t, s)
- s*2 + 5*laplace(y(t), t, s) - 4*2 -1 == (s + 3)/(s^2 + 6*s + 10)
sage: sol = solve(eq2,laplace(y(t), t, s)); show(sol)

[$\displaystyle \mathcal{L}\left(y\left(t\right), t, s\right) = \frac{2 \, s^{3} + 21 \, s^{2} + 75 \, s + 93}{s^{4} + 10 \, s^{3} + 39 \, s^{2} + 70 \, s + 50}$]
_

s^2*laplace(y(t), t, s) + 4*s*laplace(y(t), t, s) - s*y(0) + 5*laplace(y(t), t, s) - 4*y(0) - D[0](y)(0) == (s + 3)/(s^2 + 6*s + 10)
sage: t=var('t');y=function('y')(t)
sage: show(desolve_laplace(diff(y,t,2)+4*diff(y,t)+5*y==
(e^(-3*t))*cos(t),y))

$\displaystyle \frac{1}{5} \, {\left({\left(5 \, y\left(0\right) - 1\right)} \cos\left(t\right) + {\left(10 \, y\left(0\right) + 5 \, \mathrm{D}_{0}\left(y\right)\left(0\right) + 3\right)} \sin\left(t\right)\right)} e^{\left(-2 \, t\right)} + \frac{1}{5} \, {\left(\cos\left(t\right) - 2 \, \sin\left(t\right)\right)} e^{\left(-3 \, t\right)}$
sage: t=var('t'); y=function('y')(t)
sage: show(desolve_laplace(diff(y,t,2)+4*diff(y,t)+5*y==
(exp(-3*t))*cos(t),y,[0,2,1]))

$\displaystyle \frac{1}{5} \, {\left(9 \, \cos\left(t\right) + 28 \, \sin\left(t\right)\right)} e^{\left(-2 \, t\right)} + \frac{1}{5} \, {\left(\cos\left(t\right) - 2 \, \sin\left(t\right)\right)} e^{\left(-3 \, t\right)}$
sage: t,r,k=var('t,r,k')
sage: y=function('y')(t)
sage: show(desolve_laplace(diff(y,t)==r*y*(1-(y/k)),y,ivar=t))

$\displaystyle \mathcal{L}^{-1}\left(-\frac{r \mathcal{L}\left(y\left(t\right)^{2}, t, g_{13120}\right) - k y\left(0\right)}{g_{13120} k - k r}, g_{13120}, t\right)$
var('t s')

(t, s)
sage: y=function('y')(t)
sage: show((diff(y,t,2)+4*y).laplace(t,s))

$\displaystyle s^{2} \mathcal{L}\left(y\left(t\right), t, s\right) - s y\left(0\right) + 4 \, \mathcal{L}\left(y\left(t\right), t, s\right) - \mathrm{D}_{0}\left(y\right)\left(0\right)$
sage: s,t = var('s,t')
sage: g = piecewise([[(-infinity,5),0],[[5,10],(t-5)/5],[(10,infinity),1]]);g

piecewise(t|-->0 on (-oo, 5), t|-->1/5*t - 1 on [5, 10], t|-->1 on (10, +oo); t)
sage: eq = s^2*laplace(y(t), t, s) + 4*laplace(y(t), t, s)==g.laplace(t,s)
sage: show(eq)

$\displaystyle s^{2} \mathcal{L}\left(y\left(t\right), t, s\right) + 4 \, \mathcal{L}\left(y\left(t\right), t, s\right) = -\frac{{\left(5 \, s + 1\right)} e^{\left(-10 \, s\right)}}{5 \, s^{2}} + \frac{e^{\left(-10 \, s\right)}}{s} + \frac{e^{\left(-5 \, s\right)}}{5 \, s^{2}}$
show(solve(eq,laplace(y(t), t, s)))

[$\displaystyle \mathcal{L}\left(y\left(t\right), t, s\right) = \frac{{\left(e^{\left(5 \, s\right)} - 1\right)} e^{\left(-10 \, s\right)}}{5 \, {\left(s^{4} + 4 \, s^{2}\right)}}$]

sage: y=function('y')(x)
sage: sol = desolve(diff(y,x)+2*y==9,y,[0,1])
sage: fig1=plot(sol,x,[-0.5,1])
sage: f(x,y)=(1,9-2*y)
sage: fig2=plot_vector_field(f, (x,-0.6,1), (y,-5,4), color='gray')
sage: show(fig1 + fig2, aspect_ratio = 0.11, axes=False)

sage: t=var('t'); x=function('x')(t); y=function('y')(t)
sage: desolve_system([diff(x,t) == -y,diff(y,t)==x],[x,y])

[x(t) == cos(t)*x(0) - sin(t)*y(0), y(t) == sin(t)*x(0) + cos(t)*y(0)]
sage: sol=desolve_system([diff(x,t)==y,diff(y,t)==-x],[x,y],[0,1,0]);
sage: solx=sol[0].rhs();soly=sol[1].rhs()
sage: parametric_plot3d((t,solx,soly),(0,2*pi))

3D rendering not yet implemented
sage: t=var('t')
sage: x,y=function('x')(t),function('y')(t)
sage: desolve_system([diff(x,t)==-y,diff(y,t)==x],[x,y],[0,1,0])

[x(t) == cos(t), y(t) == sin(t)]
sage: sol=desolve_system([diff(x,t)==-y,diff(y,t)==x],[x,y],[0,1,0])
sage: solx=sol[0].rhs();soly=sol[1].rhs()
sage: p1 = parametric_plot((solx,soly),(0,2*pi))
sage: p2=plot(solx,t,(0,4*pi));p3 = plot(soly,t,(0,4*pi), color='green')
sage: G = graphics_array([p1,p2+p3])
sage: G.show(figsize=[10,3])




G = graphics_array([p1,p2+p3])

G.show(figsize=[10,3])

G.show(figsize=[10,3])

sage: sol=desolve_system([diff(x,t)==y,diff(y,t)==-x],[x,y],[0,1,0])
sage: solx=sol[0].rhs();soly=sol[1].rhs()
sage: parametric_plot3d((t,solx,soly),(0,4*pi),
thickness=3, aspect_ratio=(0.25,1,1))

sage: x,y=PolynomialRing(QQ,2,"xy").gens()



sage: x,y = polygens(QQ, 'x,y')
sage: eulers_method(9-2*y,0,1,1/3,2,algorithm='none')

[[0, 1], [1/3, 10/3], [2/3, 37/9], [1, 118/27], [4/3, 361/81], [5/3, 1090/243], [2, 3277/729], [7/3, 9838/2187]]
sage: pts=_
sage: fig1=list_plot(pts);fig2=line(pts, linestyle='--')
sage: t=var('t');z=function('z')(t)
sage: fig3=plot(desolve(diff(z,t)==9-2*z,z,[0,1]),t,[0,7/3])
sage: show(fig1+fig2+fig3)

desolve_laplace?

File: /projects/sage/sage-7.5/local/lib/python2.7/site-packages/sage/calculus/desolvers.py
Signature : desolve_laplace(de, dvar, ics=None, ivar=None)
Docstring :
Solve an ODE using Laplace transforms. Initial 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

AUTHORS:

* David Joyner (1-2006,8-2007)

* Robert Marik (10-2009)

sage: t = var('t')
sage: psi = function('psi',t)
sage: a = 3 * diff(psi)^2 - 2 * diff(psi) + 1
sage: show(a)


$\displaystyle 3 \, \frac{\partial}{\partial t}\psi\left(t\right)^{2} - 2 \, \frac{\partial}{\partial t}\psi\left(t\right) + 1$
sage: a.substitute(diff(psi)==t)

3*t^2 - 2*t + 1