Hi please complete the following question with full working

Hi please complete the following question with full working out

(a) Solve y\"-2y\' +2y = 0 where y (0) = 0 and y\'(n) = 1.

Solution

y\" -2y\' +y =0

Let y = u ; u\" - 2u\' +u =0

characteristic equation is : r^2 -2r +1 =0

(r -1)^2 =0 ; r= 1,1

General solution : y = c1e^x + xc2e^x

Find c1 and c2 using y(0) =1 and y\'(pi) =1

So, y(0) = 1 ; 1 = c1

y = e^x + xc2e^x

y\' = e^x + c2( xe^x + e^x)

y\'(pi) =1 ; 1 = e^pi + c2( pie^pi + e^pi)

c2 = ( 1-e^pi)/e^pi( 1+pi)

So, y = e^x + [ ( 1-e^pi)/e^pi( 1+pi)]xe^x