def picard_iteration(f, a, c, N):
    '''
    Computes the N-th Picard iterate for the IVP  

        x' = f(t,x), x(a) = c.

    EXAMPLES:
        sage: var('x t s')
        (x, t, s)
        sage: a = 0; c = 2
        sage: f = lambda t,x: 1-x
        sage: picard_iteration(f, a, c, 0)
        2
         sage: picard_iteration(f, a, c, 1)
        2 - t
        sage: picard_iteration(f, a, c, 2)
        t^2/2 - t + 2
        sage: picard_iteration(f, a, c, 3)
        -t^3/6 + t^2/2 - t + 2
        sage: var('x t s')
        (x, t, s)
        sage: a = 0; c = 2
        sage: f = lambda t,x: (x+t)^2
        sage: picard_iteration(f, a, c, 0)
        2
        sage: picard_iteration(f, a, c, 1)
        t^3/3 + 2*t^2 + 4*t + 2
        sage: picard_iteration(f, a, c, 2)
        t^7/63 + 2*t^6/9 + 22*t^5/15 + 16*t^4/3 + 11*t^3 + 10*t^2 + 4*t + 2

    '''
    if N == 0:
        return c*t**0
    if N == 1:
        #print integral(f(s,c*s**0), s, a, t)
        x0 = lambda t: c + integral(f(s,c*s**0), s, a, t)
        return expand(x0(t))
    for i in range(N):
        x_old = lambda s: picard_iteration(f, a, c, N-1).subs(t=s)
        #print x_old(s)
        x0 = lambda t: c + integral(f(s,x_old(s)), s, a, t)
    return expand(x0(t))
v=var('x t s')
a = 0; c = 4; N=10; b=2;
f = lambda t,x: sin(t)-2*x; assume(t>0) 
z=[picard_iteration(f, a, c, i) for i in range(N+1)]
for i in range(N+1):
    show(z[i])
from sage.plot.colors import rainbow
c=rainbow(N+1)
where = [x,-b+1,b]
p=plot(-1/5*(cos(t)*e^(2*t) - 2*e^(2*t)*sin(t) - 21)*e^(-2*t),where,color='gray',gridlines=True)                     #Solución exacta.
#p+=plot(z[0],where,gridlines=True)
for i in range(7,N+1):
    p+=plot(z[i],where,color=c[i])

show(p)

N=50;b=1
from sage.plot.colors import rainbow
c=rainbow(N+1)
where = [x,0,b]
p=plot(x^0,where,color='gray',gridlines=True)
for i in range(1,N+1):
    p+=plot(x^i,where,color=c[i])
show(p)

x = var('x')
y = function('y')(x)
show(desolve(diff(y,x) - exp(x+y), y))

x = var('x')
y = function('y')(x)
f = desolve(diff(y,x) -exp(x+y), y, ics=[0,1]); show(f)

t = var('t')
x = function('x')(t)
f = desolve(diff(x,t) -sin(t) + 2*x, x, ics=[0,4]); f

-1/5*(cos(t)*e^(2*t) - 2*e^(2*t)*sin(t) - 21)*e^(-2*t)

t = var('t')
x = function('x')(t)
f = desolve(diff(x,t) -sin(t) + 2*x, x, ics=[0,4]); show(f)

def picard_iteration(f, a, c, N):
    '''
    Computes the N-th Picard iterate for the IVP  

        x' = f(t,x), x(a) = c.

    EXAMPLES:
        sage: var('x t s')
        (x, t, s)
        sage: a = 0; c = 2
        sage: f = lambda t,x: 1-x
        sage: picard_iteration(f, a, c, 0)
        2
         sage: picard_iteration(f, a, c, 1)
        2 - t
        sage: picard_iteration(f, a, c, 2)
        t^2/2 - t + 2
        sage: picard_iteration(f, a, c, 3)
        -t^3/6 + t^2/2 - t + 2
        sage: var('x t s')
        (x, t, s)
        sage: a = 0; c = 2
        sage: f = lambda t,x: (x+t)^2
        sage: picard_iteration(f, a, c, 0)
        2
        sage: picard_iteration(f, a, c, 1)
        t^3/3 + 2*t^2 + 4*t + 2
        sage: picard_iteration(f, a, c, 2)
        t^7/63 + 2*t^6/9 + 22*t^5/15 + 16*t^4/3 + 11*t^3 + 10*t^2 + 4*t + 2

    '''
    if N == 0:
        return c*t**0
    if N == 1:
        #print integral(f(s,c*s**0), s, a, t)
        assume(s>0) 
        x0 = lambda t: c + integral(f(s,c*s**0), s, a, t)
        return expand(x0(t))
    for i in range(N):
        x_old = lambda s: picard_iteration(f, a, c, N-1).subs(t=s)
        #print x_old(s)
        x0 = lambda t: c + integral(f(s,x_old(s)), s, a, t)
    return expand(x0(t))
v=var('x t s')
a = 0; c = 1; N=2; b=.5;
f = lambda t,x: exp(x+t); assume(t>0) 
z=[picard_iteration(f, a, c, i) for i in range(N+1)]
for i in range(N+1):
    show(z[i])
from sage.plot.colors import rainbow
c=rainbow(N+1)
where = [x,-b,b]
p=plot(-log(abs(-1-exp(-1)+exp(t))),where,color='gray',gridlines=True)                     #Solución exacta.
#p+=plot(z[0],where,gridlines=True)
for i in range(N+1):
    p+=plot(z[i],where,color=c[i])

show(p)