1D Finite Difference Method

Hello guys, I'm very new at Matlab. Currently i'm working the following task:

given ODE:

with the following conditions:

and ,

Analytic solution:

Task: Implement an iterative Finite Difference scheme based on backward, forward and central differencing to solve this ODE. Use an array to store the N unknowns (DOFs) . Implement the scheme in a function of the time step width which returns the DOF array as result. Plot the solution vector for a time step width

of ∆? = 0.1 ? together with the analytic solution in the same graph. Select N large enough to ensure that the simulation will at least simulate the solution for the first

second.

I'm trying to get the backward differencing first:

%Main Program
close all;
clear all; 

dt=0.1;
N=round(1/dt+0.5);

backwardDiff(N,dt);
analyticSolution(N,dt);


function backwardDiff(N,dt)

dof=zeros(1,N);
g = -9.8;

%to get rid of negative degrees of freedom
dof(1) = 4;
dof(2) = g*dt+dof(1);

for i=3:N
    dof(i) = g*dt*dt+2*dof(i-1)-dof(i-2)
end


%fill time vector with spacing 0.1 for the plot
startValue = 0.0;
endValue = 1.0;
Elements = 11;
stepSize=(endValue-startValue)/(Elements-1);
timeVector = startValue:stepSize:endValue 

plot(timeVector,dof)


end

%analytical Solution
function analyticSolution(N,dt)

%fill time vector with spacing 0.1 for the plot
startValue = 0.0;
endValue = 1.0;
Elements = 11;
stepSize=(endValue-startValue)/(Elements-1);
timeVector = startValue:stepSize:endValue; 

sVector = zeros(1,N);

for i=1:N
    sVector(i) = 4+0.5*(-9.8)*timeVector(i)^2
end

plot(timeVector, sVector)
ylabel("S/t")
xlabel("t")

I have no clue how to proceed with the central and forward method. Moreover how do I place all the results on the same graph?

Thank you.