I'd like to choose one solution technique and write the finite difference form of the solution. Can you please explain how that leads to the next estimate(s), the accuracy of the method, and why you chose that method. I'd use only each technique once?

Methods are Backward, Forward Euler, Crank–Nicolson, Runge–Kutta or Adams–Bashforth. I tried to find an analytical solution for each first, then compare the results with the exact results of the point for each time step. However, solving the equations are so difficult. Is there any way to determine a method without solving the equations?

df/dt=f^{^(1/3)}(1/4t)

df/dt=2*t

df/dt=−e^^(f*t)

df/dt=f(∂2f/∂x2)

1) This looks like homework/coursework. Is it?

2) You have not specified the domains ie what are the boundary/initial conditions?

3) Is equation 3

[math]\frac{{df}}{{dt}} =  - {e^{ft}}[/math]

or

[math]\frac{{df}}{{dt}} =  - {e^{f\left( t \right)}}[/math]

 

4) Why is  equation 2 difficult analytically?

[math]\frac{{df}}{{dt}} = 2t[/math]

Separate variables yields

df = 2tdt and integrate

 

Note this site provides useful superscript/subscript.
You can also use exp (..) for exponentials