1. Implement the Explicit Euler Scheme for Initial Value Problems of the form:
	y'(t) = F(t, y(t)) ,  t0 ≤ t ≤ tend
	y(t0) = y0

   The function F(t,y) should be coded in a function subprogram FCN(...).
   Input data: t0, y0, tend, Nsteps.  Thus the time-step will be h=(tend-t0)/Nsteps.
   Your code should print out the input data and then the pairs:
		tn          Yn
   At the end, it should print out the final n, tn, Yn
   (appropriately labelled, of cource).

2. Solve, on paper, the (exponential growth) problem:  y' = y ,  0 ≤ t ≤ 1 ,   y(0) = −1

3. To debug your code, run it on the problem above. 
   Compare the numerical solution Yn with the exact solution yEXACT(tn),
   i.e. modify your output to print out:
	tn	Yn	yEXACTn		ERRn
   where ERRn = |Yn - yEXACT(tn)|, and keep track of the maximum error.
   At the end of the run, print out the above values (at time tend) 
   and the maximum overall error ERRmax.  Test with N=10 and N=100.
   Turn off printing of tn  Yn ... and test with N = 1000, 10000 and larger.

   Once the code is debugged, only FCN(...) and input data need to be changed to solve other IVPs.
  

4. Now solve the IVP:
	y' = −t/y ,   0 ≤ t ≤ 1
        y(0) = 1

   Find the exact solution at t = 1 (by hand),
   and compare the numerical and exact values at t = 1.  
   Try small (N=10) and larger (N=1000, 100000, ... ) number of time-steps.
   At which time does the worst error occur in this problem ?
   Plot the exact solution.  Do you see why it occurs there?
 

5. Solve it using a Matlab / Octave / Python solver, each contains several:
      Matlab: ode45  ode78  ode113 , and for stiff: ode15s ode23s  ...
      Octave: ode45  lsode  ode15s ...
      Python: scipy.integrate.solve_ivp , diffeqpy ...
      Syntax for Matlab / Octave: ode45( @FCN, t_range, y0 )
      e.g for the IVP in 2.:  [t, y] = ode45( @(t,y) y , [0,1], -1 )

   Try ode45 (adaptive 4th order) and some other(s) on the IVP in 4.


Submit, in a plain text file  Lab7.txt:
 a. output from 3. for N=10.
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 b. result from 4. for N=100000 (only the final result!!!)
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 c. How many correct digits did you get ?  Is there a reason for the large errors ? 
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 d. result from 5: show command you used and solution for t=1.
    Did any do better than in b. ?
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 e. your Euler code, including FCN for problem in 4.