$$y''+6y'+9y=1+x$$
$$r_1=r_2=-3$$
$$y_c=c_1e^{-3x}+c_2xe^{-3x}$$
$$y_p(x)=Ax+B$$
$$y_p'(x)=A$$
$$y_p''(x)=0$$
$$0+(9A)x+(6A+B)=1+x$$
$$A=\frac19$$
$$B=\frac13$$
$$y_p(x)=\frac19x+\frac13$$
$$y(x)=c_1e^{-3x}+c_2e^{-3x}+\frac19x+\frac13$$
how did I do?

Question

$$y''+6y'+9y=1+x$$
$$r_1=r_2=-3$$
$$y_c=c_1e^{-3x}+c_2xe^{-3x}$$
$$y_p(x)=Ax+B$$
$$y_p'(x)=A$$
$$y_p''(x)=0$$
$$0+(9A)x+(6A+B)=1+x$$
$$A=\frac19$$
$$B=\frac13$$
$$y_p(x)=\frac19x+\frac13$$
$$y(x)=c_1e^{-3x}+c_2e^{-3x}+\frac19x+\frac13$$
how did I do?

dumbcow · Answer

pretty good...i only see 1 thing

it should be 0+(9A)x +(6A+9B) = 1+x
9(Ax +B) = (9A)x + 9B

it only changes the B value to 1/27

anonymous · Answer

oh thank you! silly me....hihi