Find a particular solution?

Question

Use the method of variation of parameters to find a particular solution of y"-5y'+6y=2e^t.

Answer: Y(t)=e^t

r=2, 3

Answer 1

A particular solution is evident. If you have exponential function  on the right side of the equation yuo can look for a particular solution in the form:

                                                       y = A e^t

since all derivatives give you the same function, Substitute this function into the left side of the eqaution.

It gives you (after dividing by e^t)):

                                          A - 5A + 6A = 2    or    A = 1      

Thus     y = e^t .     

You don't need to use r =2,3. This is about general solution for the homogeneous equation.