When solving an equation in the form $$ ay''+by'= cx^2+dx+f $$ I find that it has a complementary equation with a constant, and so I have difficulties when I attempt to find the particular integral.
Usually the particular integral would be of the form $y=Kx^2+Lx+M$: however since the complementary equation has a constant my maths book tells me to use $y= Kx^3+Lx^2+Mx$ and I don't understand why. Why does a complementary equation with a constant term influence the form of the particular integral?
$\lambda=0$ is solution of characteristic equation $a\lambda^2+b\lambda=0$ order $s=1$,
$cx^2+dx+f$ is polynomial degree $2$. Then $$y_p=x^s(Kx^2+Lx+N)=x(Kx^2+Lx+N).$$
You can get $$y_p=x\, \left( \frac{c\, {{x}^{2}}}{3 b}+\frac{\left( b d-2 a c\right) x}{2 {{b}^{2}}}+\frac{{{b}^{2}} f-a b d+2 {{a}^{2}} c}{{{b}^{3}}}\right) $$