FORMULAS APPROXIMATING DERIVATIVES OF A FUNCTION f(x)

Fisrt Forward-Differentiation Formulas:

Fisrt Central-Differentiation Formulas:

Fisrt Backward-Differentiation Formulas:

Second Forward-Differentiation Formulas:

Second Central-Differentiation Formulas:

Second Backward-Differentiation Formulas:

**Euler's Method:** x_{k+1} = x_{k}+h

y_{k+1} = y_{k}+hf(x_{k} , y_{k}) for k=0,1,2,3,……..

**Heun's Method:** x_{k+1} = x_{k}+h

for k=0,1,2,3,……..

**Backward Euler's Method: **x_{k+1} = x_{k}+h

y_{k+1} = y_{k}+hf(x_{k+1} , y_{k+1}) for k=0,1,2,3,……..

Fourth-Order Runge-Kutta Method:

, ,

,

Shooting Method:

Third guess =

first guess+(second guess-first guess) / (result of second guess-result of first guess)* (true value -result of first guess)