I am presented with the differential equation
Finding the complementary function was OK. I obtained solutions to the characteristic equation of $\lambda = 0, \pm 1, \pm i$ so the complementary function is
Now for the particular integral, my first thought would be to use a trial form
However this clearly cannot be the solution as, plugging it in, I get $-b_1=x$. Trying $b_2x^2+b_1x+b_0$ I get $-2b_2x-b_1=x$ which works for $b_1=0, b_2=-1/2$ and I suppose the constant $b_0$ doesn't really matter because it is amalgamated into the constant in the complementary function anyway.
However I am finding it quite strange that the form of the particular integral was $b_2x^2+b_1x+b_0$, and not the general form for a linear forcing term of $b_1x+b_0$ that I have been taught to use.
I was wondering if there is a rule particular integrals when you have different orders of ODE? I can see here that my first trial would fail because there is no $y$ term on the RHS. So I suppose this trial form would also in fact fail for a second order ODE of form $ay''+by'=x$. Do you just have to make observations like this in guessing the particular integral? I have never come across this before in class.