Hi I am studying 'Explicit Methods Based on Quadrature'.

I found that 'Heun's method may refer to the improved or modified Euler's method (that is, the explicit trapezoidal rule)'. See here for more details.

From what I understood the idea of the explicit trapezoidal rule is to replace the definite integral on the right-hand side with a numerical approximation.

Where the formula for the main Trapezium rule for a specific differential equation is:

$$ Y_{n+1}=Y_n+\frac{h}{2} \bigg[f(t_n,Y_n)+f(t_{n+1},y_{n+1}) \bigg]+ \mathcal{O}(h^3)$$

So, we are writing the formula $Y_{n+1}$ in terms of $Y_n, t_n $ and $t_{n+1}$.

If I have the function:

$$f(t,y)= \frac{8 \cos(4t)+y}{4}$$

and want to express the above explicit formula that uses the specific RHS of the function $f(t,y)$

Should I express it as following:

$$Y_{n+1}=Y_n+\frac{h}{2} \bigg[\frac{8 \cos(4t_n)+y_n}{4}+\frac{8 \cos(4t_{n+1})+y_{n+1}}{4} \bigg]+\mathcal{O}(h^3)$$

At each step the unknown is $Y_{n+1}$ I am trying to solve for it in the above equation as an explicit function of $Y_n $ but I am not sure how to do it.

Can anyone help me on this?

Thank you