I'm currently taking a differential equations course and we've had a few quizzes and exams where I'm solving a High Order Differential Equation (non-homogeneous) and I have to stop mid-problem because what I'm currently doing will not work. (I'm talking about, for example, if the equations is $$ y''''+2y'''+2y''=3x^2 $$ then I will get the complement solution, no biggie, but when I try to get the particular solution, I do $(Ax^2 + Bx + C)$ , and after attempting to solve it, we know it won't work. So then I try $(Ax^3 + Bx^2 + C)$, and after doing the work again, STILL IT DOESN'T WORK! Not until I do $(Ax^4 + Bx^3 + Cx^2 + Dx + E)$ Does it work, BUT AFTER ALL THAT TIME I'VE LOST LIKE 7 MINUTES OR MORE AND MY EXAM/QUIZZES SCORES ARE AFFECTED.
Is there a way of knowing ahead of time to start with $Ax^4....$ rather than $Ax^2$ ?
The lowest order derivative in $y'''' + 2y''' + 2y''$ is second-order. That means any $n$th degree polynomial for $y$ will result in an at most $n-2$th degree polynomial. Since your inhomogeneous term is degree 2, you need at least a fourth degree polynomial for your trial function.
In general, if your differential operator has constant coefficients and lowest order $n$, and your inhomogeneous term is a polynomial of degree $m$, your trial function will need to be a polynomial of degree $n+m$.