Question

y'' -5y'+6y= x

Answer 1

complementary solution is m^2-5m+6=0 (m-3) (m-2) = 0 so
yc=c1e^(2x)+c1e^(3x)

particular solution let yp=Ax+B find A and B by plugging into the Diff. Eq.
yp'=A
yp''=0

y''-5y'+6y=x
0-5A+6Ax+6B=x so
6A=1, A= 1/6
-5A+6B=0, B=(5/6)A=(5/6)(1/6)=5/36

so yp=(1/6)x+(5/36)

the entire solution is y=yc+yp=c1e^(2x)+c1e^(3x)+(1/6)x+(5/36)