A particular solution of the equation y y et sin t should

A particular solution of the equation y\"\' -y = e^t + sin t should have the form ae^t + b sin(t). ate^t + be^t + c sin^t. ate^t + b sin^t + c cos t. ate^t + bt cos (t) + ct sin t.

Solution

e^t is solution to homogeneous equation so the guess from e^t is

at e^t

From sin t guess is

b sin t+ c cos t

So the guess is

a t e^t +b sin t +c cos t