Using Euler's method

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You problem is not clear, I need to guess, I think your problem is as follows, if not, please clarify.

Problem:

y'= 1/2 - y+2x when y(0)=1
Find the exact solution of y(x)
Let h=.1 use euler & improved to appoximate to get
"Phee" of .1, phee of.2, and phee of .3

Solution:

We write equation of the form y'=-y+1/2+2x and

First we solve y'= - y

We know dy/(-y)=dx, integrate both sides to get

-ln|-y|=x+C.

Thus

-y=+e^(-x-C)=+e^(-C)e^(-x) or
-y=-e^(-x-C)=-e^(-C)e^(-x)

Thus

y=ke^(-x) for some constant k

Now we solve

y'= 1/2 - y+2x, we expect solutions to be of the form

y=f(x)(e^(-x)), for ...