1214.sagews

%latex
$$
\left\lbrace\begin{array}{rcl}
y'(t) &=& 1 + \frac{y}{t} + \left( \frac{y}{t}\right)^2\\
y(1) &=& 0
\end{array}\right.$$

The exact solution of the above IVP is given by $y(t) = t \tan(\ln t) $

a)Obtain numerical solution for the IVP given above by using Euler’s method,
Midpoint method (Runge-Kutta method order 2), Modified Euler’s method, and
Runge-Kutta method order 4. Use h = 0.01 and calculate the error for each
method at each point. Organize your results into a table or tables.

%latex
Euler's Method Algorithm is as follows

def eulers_method(a,b,N,alpha):
    h = (b-a)/N
    t = a
    f = lambda x,y: 1.0 + (y/x) + (y/x)^2
    omega = alpha
    #for storage of values, we create a list
    w = []
    w.append((0,omega))
    for i in range(1,N):
        omega = omega + h * f(t,omega)
        w.append((i,omega))
        t = a + i * h
    return w

#since h = 0.01, N = (4-1)/.01 = 300
ans = eulers_method(1,4,300,0) # returns a table of points (i, w_i)
mat_ans = matrix(ans)# creates a matrix of ans
Latex_Table = latex(mat_ans)#makes the matrix into a table array in Latex

%latex
The table is so large I cannot display it in the notebook.  I will show you the problem in Wave.  I will have to truncate the table. :-(