from sage.calculus.desolvers import desolve_odeint
x,y=var('x,y')
delta=-1
tau=0
init=[0.5,1]
sr=srange(0,0.5,0.1)
f=[-delta*y,x+tau*y]
p1=plot_vector_field(f,(x,-2,2),(y,-2,2))
sol=desolve_odeint(f,init,sr,[x,y])
p2=line(zip(sol[:,0],sol[:,1]))
show(p1+p2)
p=p1+p2
#p.save('zadel.eps')
g=function('g')(x)
soll=desolve(diff(g,x,2)-tau*diff(g,x,1)+delta*g==0,g,[0,init[0],init[1]]).simplify_full()
print(expand(soll))
line(zip(sr,sol[:,0]))+line(zip(sr,sol[:,1]))
-1/4*e^(-x) + 3/4*e^x -1/4*e^(-x) + 3/4*e^x  
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from sage.calculus.desolvers import desolve_odeint
x,y=var('x,y')
delta=1
tau=0
init=[0.5,1]
sr=srange(0,10,0.1)
f=[-delta*y,x+tau*y]
p1=plot_vector_field(f,(x,-2,2),(y,-2,2))
sol=desolve_odeint(f,init,sr,[x,y])
p2=line(zip(sol[:,0],sol[:,1]))
show(p1+p2)
p=p1+p2
#p.save('centra.eps')
g=function('g')(x)
soll=desolve(diff(g,x,2)-tau*diff(g,x,1)+delta*g==0,g,[0,init[0],init[1]]).simplify_full()
print(expand(soll))
line(zip(sr,sol[:,0]))+line(zip(sr,sol[:,1]))
1/2*cos(x) + sin(x) 1/2*cos(x) + sin(x)  
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from sage.calculus.desolvers import desolve_odeint
x,y=var('x,y')
delta=1
tau=1
init=[-0.1,0.1]
sr=srange(0,6,0.1)
f=[-delta*y,x+tau*y]
p1=plot_vector_field(f,(x,-2,2),(y,-2,2))
sol=desolve_odeint(f,init,sr,[x,y])
p2=line(zip(sol[:,0],sol[:,1]))
show(p1+p2)
p=p1+p2
#p.save('instabielbrandpunt.eps')
g=function('g')(x)
soll=desolve(diff(g,x,2)-tau*diff(g,x,1)+delta*g==0,g,[0,init[0],init[1]]).simplify_full()
print(expand(soll))
line(zip(sr,sol[:,0]))+line(zip(sr,sol[:,1]))
1/10*sqrt(3)*e^(1/2*x)*sin(1/2*sqrt(3)*x) - 1/10*cos(1/2*sqrt(3)*x)*e^(1/2*x) 1/10*sqrt(3)*e^(1/2*x)*sin(1/2*sqrt(3)*x) - 1/10*cos(1/2*sqrt(3)*x)*e^(1/2*x)  
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from sage.calculus.desolvers import desolve_odeint
x,y=var('x,y')
delta=1
tau=-1
init=[-1,1]
sr=srange(0,10,0.1)
f=[-delta*y,x+tau*y]
p1=plot_vector_field(f,(x,-2,2),(y,-2,2))
sol=desolve_odeint(f,init,sr,[x,y])
p2=line(zip(sol[:,0],sol[:,1]))
show(p1+p2)
p=p1+p2
#p.save('stabielbrandpunt.eps')
g=function('g')(x)
soll=desolve(diff(g,x,2)-tau*diff(g,x,1)+delta*g==0,g,[0,init[0],init[1]]).simplify_full()
print(expand(soll))
line(zip(sr,sol[:,0]))+line(zip(sr,sol[:,1]))
1/3*sqrt(3)*e^(-1/2*x)*sin(1/2*sqrt(3)*x) - cos(1/2*sqrt(3)*x)*e^(-1/2*x) 1/3*sqrt(3)*e^(-1/2*x)*sin(1/2*sqrt(3)*x) - cos(1/2*sqrt(3)*x)*e^(-1/2*x)  
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from sage.calculus.desolvers import desolve_odeint
x,y=var('x,y')
delta=1
tau=4
init=[-0.1,0.1]
sr=srange(0,1,0.1)
f=[-delta*y,x+tau*y]
p1=plot_vector_field(f,(x,-2,2),(y,-2,2))
sol=desolve_odeint(f,init,sr,[x,y])
p2=line(zip(sol[:,0],sol[:,1]))
show(p1+p2)
p=p1+p2
#p.save('instabielknooppunt.eps')
g=function('g')(x)
soll=desolve(diff(g,x,2)-tau*diff(g,x,1)+delta*g==0,g,[0,init[0],init[1]]).simplify_full()
print(soll)
line(zip(sr,sol[:,0]))+line(zip(sr,sol[:,1]))
1/20*((sqrt(3) - 1)*e^(2*sqrt(3)*x + 2*x) - (sqrt(3) + 1)*e^(2*x))*e^(-sqrt(3)*x) 1/20*((sqrt(3) - 1)*e^(2*sqrt(3)*x + 2*x) - (sqrt(3) + 1)*e^(2*x))*e^(-sqrt(3)*x)  
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from sage.calculus.desolvers import desolve_odeint
x,y=var('x,y')
delta=1
tau=-4
init=[-1,1]
sr=srange(0,20,0.1)
f=[-delta*y,x+tau*y]
p1=plot_vector_field(f,(x,-2,2),(y,-2,2))
sol=desolve_odeint(f,init,sr,[x,y])
p2=line(zip(sol[:,0],sol[:,1]))
show(p1+p2)
p=p1+p2
#p.save('stabielknooppunt.eps')
g=function('g')(x)
soll=desolve(diff(g,x,2)-tau*diff(g,x,1)+delta*g==0,g,[0,init[0],init[1]]).simplify_full()
print(expand(soll))
line(zip(sr,sol[:,0]))+line(zip(sr,sol[:,1]))
-1/6*sqrt(3)*e^(sqrt(3)*x - 2*x) + 1/6*sqrt(3)*e^(-sqrt(3)*x - 2*x) - 1/2*e^(sqrt(3)*x - 2*x) - 1/2*e^(-sqrt(3)*x - 2*x) -1/6*sqrt(3)*e^(sqrt(3)*x - 2*x) + 1/6*sqrt(3)*e^(-sqrt(3)*x - 2*x) - 1/2*e^(sqrt(3)*x - 2*x) - 1/2*e^(-sqrt(3)*x - 2*x)  
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from sage.calculus.desolvers import desolve_odeint
x,y=var('x,y')
delta=1
tau=2
init=[-0.1,0.05]
sr=srange(0,3,0.1)
f=[-delta*y,x+tau*y]
p1=plot_vector_field(f,(x,-2,2),(y,-2,2))
sol=desolve_odeint(f,init,sr,[x,y])
p2=line(zip(sol[:,0],sol[:,1]))
show(p1+p2)
p=p1+p2
#p.save('grensinstabielbrandpuntknooppunt.eps')
g=function('g')(x)
soll=desolve(diff(g,x,2)-tau*diff(g,x,1)+delta*g==0,g,[0,init[0],init[1]]).simplify_full()
print(expand(soll))
line(zip(sr,sol[:,0]))+line(zip(sr,sol[:,1]))
3/20*x*e^x - 1/10*e^x 3/20*x*e^x - 1/10*e^x  
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from sage.calculus.desolvers import desolve_odeint
x,y=var('x,y')
delta=0
tau=1
init=[-0.4,0.5]
sr=srange(0,3,0.1)
f=[-delta*y,x+tau*y]
p1=plot_vector_field(f,(x,-2,2),(y,-2,2))
sol=desolve_odeint(f,init,sr,[x,y])
p2=line(zip(sol[:,0],sol[:,1]))
show(p1+p2)
p=p1+p2
#p.save('grenszadelinstabielknooppunt.eps')
g=function('g')(x)
soll=desolve(diff(g,x,2)-tau*diff(g,x,1)+delta*g==0,g,[0,init[0],init[1]]).simplify_full()
print(expand(soll))
line(zip(sr,sol[:,0]))+line(zip(sr,sol[:,1]))
1/2*e^x - 9/10 1/2*e^x - 9/10  
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from sage.calculus.desolvers import desolve_odeint
x,y=var('x,y')
delta=0
tau=-1
init=[-0.4,0.5]
sr=srange(0,3,0.1)
f=[-delta*y,x+tau*y]
p1=plot_vector_field(f,(x,-2,2),(y,-2,2))
sol=desolve_odeint(f,init,sr,[x,y])
p2=line(zip(sol[:,0],sol[:,1]))
show(p1+p2)
p=p1+p2
#p.save('grenszadelstabielknooppunt.eps')
g=function('g')(x)
soll=desolve(diff(g,x,2)-tau*diff(g,x,1)+delta*g==0,g,[0,init[0],init[1]]).simplify_full()
print(expand(soll))
line(zip(sr,sol[:,0]))+line(zip(sr,sol[:,1]))
-1/2*e^(-x) + 1/10 -1/2*e^(-x) + 1/10  
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