Use Euler’s method with step size 0.2 to estimate y(1), where y(t) is the solution of the initial-value problem y' = 1 − xy, y(0) = 0.

Answer 1

Nothing wrong with Roger's answer, but I'd like to describe how to do it in Excel:

In a blank worksheet, in row 1, label column A with "x", column B with "y", column C with "dx" (for "delta x") and column D with "dy" (for delta "y").

In row 2, enter 0 in the "x" column, 0 in the "y" column, 0.2 in the "dx" column. So y(0)=0 and x increments by 0.2 every step.

In the "dy" column of row 2, we enter the Excel formula =C2*(1-A2*B2), which makes dy=dx(1-xy), in this row, so dy/dx (that is delta y over delta x) is 1-xy.

Each row will have x and y increment by dx and dy, respectively in the next row.

So in row 3,

in the "x" column, we put the Excel formula =A2+C2, which is x+dx for this row;

in the "y" column, we put the Excel formula =B2+D2, which is y+dy for this row*;

* Note that when we put the formula =B2+D2 in this cell, D2 was empty. That's fixed when we fill in the value of dy for row 2.

in the "dx" column, we put the Excel formula =C2, which evaluates to 0.2 ;

in the "dy" column, we put the Excel formula =C3*(1-A3*B3), which makes dy=dx(1-xy) (so dy/dx is x(1-xy)) in this row.

In this way, the approximated values of x, y, dx, and dy from row 2 are used to compute x and dx, and approximations for y and dy, in row 3.

Then, copy row 3, columns A through D, to the range A4:D7** .

** I used my fingers to figure out that row 7 would have x=1.

The y value next to each x value is the Euler approximation of y(x), in your notation.

The values that your resulting worksheet show (with y(1)=0.782425 at the bottom of the "y" column):