From what I understand of Milne's Method, there is insufficient information to establish the predictors and correctors. However, the DE is solvable.
Multiply y'=x+y by e-x:
e-xy'=xe-x+ye-x,
e-xy'-ye-x=xe-x,
d(e-xy)/dx=xe-x.
e-xy=∫xe-xdx.
Let u=x, then du=dx; let dv=e-xdx, v=-e-x.
∫xe-xdx=-xe-x+∫e-xdx=-xe-x-e-x.
Therefore e-xy=-xe-x-e-x+C, where C is the integration constant.
Given that y(0)=1, e0=0-1+C, C=2.
e-xy=-xe-x-e-x+2 can be written y=-x-1+2ex.
CHECK
dy/dx=-1+2ex; but -1+2ex=x+y, so dy/dx=y'=x+y.
With this solution it is now possible to create a table of values which provide the necessary information to use Milne's Method.
x |
y |
0 |
1.0000 |
0.1 |
1.1103 |
0.2 |
1.2428 |
0.3 |
1.3997 |
0.4 |
1.5836 |