# Solving Differential Equations faster

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 $Ax^2$ ?

• I'd start with the exponent corresponding to the highest order differential. If you get zeros, you find out quickly anyway. – user121330 Jun 27 '17 at 22:10
• The equation is linear. If you have computed the left hand side for $Ax^2+Bx+C$ then the left hand side for $Dx^3 + Ax^2 + Bx + C$ is found by simply computing it for $Dx^3$ and adding it to the result, you don't have to recompute all the terms. Also to see why $x^4$: look at the equation and note that the smallest derivative on the left hand side is $y''$ and if this is supposed to give $x^2$ then $y$ has to contain at least $x^4$ as a derivative on a power reduces the exponent by $1$. – Winther Jun 27 '17 at 22:11

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 $n+m$.