Originally Posted by

**sixstringartist** Hello,

I am currently working on solving nonhomogeneous differential equations using the method of undetermined coefficients. Im doing well solving problems, but I have come across one that has puzzled me, possibly due to its simplicity.

The initial value problem is:

$\displaystyle y'' + y' - 2y = 2t, ~ ~ y(0) = 0, ~ ~ y'(0) = 1$

I have found the complementary general solution,

$\displaystyle y(t) = c_1e^t + c_2e^{-2t}$

but before I can solve for the constants, I must account for the nonhomogeneous term.

I make my first assumption that the term is of the form:

$\displaystyle Y(t) = At $

therefore

$\displaystyle Y'(t) = A, $

$\displaystyle Y''(t) = 0 $

$\displaystyle

A - 2At = 2t $

Obviously this is not a solution so I try the only other thing I know to try.

$\displaystyle Y(t) = At^2 $

$\displaystyle \therefore Y'(t) = 2At,$

$\displaystyle \ Y''(t) = 2A $

so

$\displaystyle 2A + 2At - 2At^2 = 2t$

This isnt going to work either and now Im out of ideas. Any help is greatly appreciated.