import math
def eule(f,a,b,xa,n):
  t = a
  x = xa
  h = (b-a)/n
  while t < b :
    x += h * f(x,t)
    t += h
  print('the approx solu is x =', b,'is',x)

#3a
def f(x,t):
  return x
eule(f, 0, 1, 1, 1)
eule(f, 0, 1, 1, 2)
print('\n')


#3b
def eule1(f,a,b,xa,n):
  t = a
  x = xa
  h = (b-a)/n
  while t < b:
    x += h * f(x,t)
    t += h
  print(' the approx solu is x =', b,'is',x,'and the abs error is',2.41287593875817 - x)

def g(x,t):
  return t + math.sin(x)
eule1(g, 0, 1, 1, 5)
eule1(g, 0, 1, 1, 25)
eule1(g, 0, 1, 1, 125)
eule1(g, 0, 1, 1, 625)
print('\n')

# 3c
def y(x,t):
  return .001*x*(900-x)
eule(y, 0, 1, 200, 100)
print('\n')

def tay2de(f, fp, a, b, xa, n):
  t = a
  x = xa
  h = (b-a)/n
  while t< b:
    x += h *f(x,t) + ((h**2)/2) * fp(x,t)
    t += h
  print(' the approx value of sol at x =',b,'is',x)