# Using Euler's method

Y'= 1/2 - (X)+2x when y(0)=1
Find the exact solution of ___ O/ <<note!!!! I don't know how to put in a zero with a line going across to make a pheee.
1a. Let h=.1 use euler & improved to approximate to get
"Phee" of .1, phee of.2, and phee of .3

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#### Solution Preview

...ome function f(x)

From y'= - y+1/2+2x we have

f'(x) e^(-x)+f(x)(- e^(-x))=- f(x)(e^(-x))+1/2+2x Thus.

f'(x)=(1/2+2x)/ e^(-x)= (1/2+2x)e^(x) =(1/2) e^(x)+2xe^(x) Now we integrate both sides

Use integration by parts formula, let u=x, v=e^(x), we have the integration of

xe^(x) is equal to xe^(x)- e^(x) . thus

f(x)=(1/2) e^(x) +2(xe^(x)- e^(x))+C=2xe^(x)-(3/2) e^(x)+C. Thus the general solution is

y=f(x)(e^(-x))= (2xe^(x)-(3/2) e^(x)+C) (e^(-x))=2x-3/2+C(e^(-x)). With y(0)=1, we

have C=5/2. Thus the solution for the exact solution for the given problem is

y=2x-3/2+(5/2)(e^(-x)).

The exact solutions at x=0.1, 0.2 , 0.3,..
are ...