What should I try for the particular integral inorder to solve these problems where D means d/dx.
Are you familiar with the so-called "Annihilator" approach?? To see how you do it this way, look at post #7 in my differential equations tutorial [also look at post #6 and #8 for two other techniques].
Going along with this, we solve the homogeneous solution first:
You should end up with
Now for the particular solution, solve the non homogeneous equation.
This will lead us to finding the annihilator of , which is
The DE becomes with multiplicity six. Thus, the particular solution with only consist of r repeating 4 times, since 2 were used in the homogeneous solution.
Thus,
Now substitute into the original DE and find the coefficients...
If you still do the annihilator approach,
Try to do something similar to what I have done above.
I hope this helps.
--Chris