Find all values of A and B, if any, such that y1 = Axe-x and y2 = Be-x are solutions of the differential equation 2y\' + 2y = e-x Solution if A is solution then put it in eq 2(-Axe-x + Ae-x ) + 2(Axe-x ) = e-x -2Ax + 2A + 2Ax = 1 2A = 1 A = 1/2 put Be-x in eq 2(-Be-x ) + 2(Be-x ) = e-x 0 = 1 can never be a solution for all values of B Be-x is never a solution.

1. Find all values of A and B, if any, such that y1 = Axe-x and y2 = Be-x are solutions of the
differential equation 2y' + 2y = e-x
Solution
if A is solution then put it in eq
2(-Axe-x + Ae-x ) + 2(Axe-x ) = e-x
-2Ax + 2A + 2Ax = 1
2A = 1
A = 1/2
put Be-x in eq
2(-Be-x ) + 2(Be-x ) = e-x
0 = 1
can never be a solution for all values of B
Be-x is never a solution