Exercise 106

Undetermined coefficients can sometimes be used to guess a particular solution to other equations than those with constant coefficients. Find a polynomial $y(x)$ that solves $y'+ x y = x^3+2x^2+5x+2$.

Note: Not every right-hand side will allow a polynomial solution, for example, $y'+xy=1$ does not, but a technique based on undetermined coefficients does work, see the “Power series methods” chapter.