Suppose the functions y1t e2t and y2t e2t span the solutio

Suppose the functions y_1(t) = e^-2t and y_2(t) = e^2t span the solution space for the homogeneous linear DE: L(y) = 0. Specify the form of the solution to the non-homogeneous equation L(y) = e^2t.

Solution

The right hand side of the non homogeneous problem is in the span of a

one-dimensional homogeneous solution subspace e^2t, corresponding to a root r= 2

of multiplicity one. Hence, the form of the particular solution is y_p=Ate^2t

.