For y "+py'+q = f (x) e (enter *x)
Enter =0 is not a characteristic root, so let the special solution be
y=ax+b
Y'=a y"=0 into the original equation.
0-2a-2(ax+b)=3x+ 1
-2ax-2(a+b)=3x+ 1
-2a=3 -2(a+b)= 1
a=-3/2
b= 1
Special solution y=-3x/2+ 1
The general solution y = c1e (3x)+c2e (-x)-3x/2+1.
For f (x) e (converted to x),
f(x)=a 1x^(nx)+a2x^[(n- 1)x]+...+a(n+ 1)
When in is one of the characteristic roots, let the special solution be
Y = xq (x) e (enter x) q (x) = a1x (NX)+...+a (n+1)
If it is not a characteristic root, let y = q (x) e (into x)