Solve the differential equation using the method of undetermined coefficients.
$$y ^ { tt } + 6 y ^ { t } + 9 y = 2 + x$$
Select the correct answer.
A) $y ( x ) = c _ { 1 } x e ^ { - 3 x } + c _ { 2 } x ^ { 2 } e ^ { - 3 x } + \frac { 4 } { 27 } x$
B) $y ( x ) = c _ { 1 } e ^ { - 3 x } + c _ { 2 } x e ^ { - 3 x } + \frac { 1 } { 9 } x + \frac { 4 } { 27 }$
C) $y ( x ) = c _ { 1 } e ^ { - 3 x } + c _ { 2 } x e ^ { - 3 x }$
D) $y ( x ) = c _ { 1 } e ^ { - 3 x } + \frac { 1 } { 9 } x + \frac { 4 } { 27 }$
E) $y ( x ) = c _ { 1 } e ^ { - 3 x } + c _ { 2 } x e ^ { - 3 x } + \frac { 2 } { 9 } x$

Question

Quizplus · Accepted Answer

The characteristic equation of $y ^ { tt } + 6 y ^ { t } + 9 y = 0$ is $r^2+6r+9=(r+3)^2=0$. Thus, the complementary solution is $y_c(x)= c_1 e^{-3x}+c_2xe^{-3x}$. Since the right-hand side of the given equation is of the form $2+x$, we assume the particular solution is of the form $y_p(x)=Ax+Bx+C$. Then, we have
$$y_p'(x)=A+B,\quad y_p''(x)=0,$$
and we substitute them into the original equation and obtain
$$(A+B)+6(A+B+C)+9(Ax+Bx+C)=2+x.$$
By equating the coefficients of the same order of $x$, we obtain $9A=1$, $6A+9B=0$, and $A+6B+9C=2$. Therefore, $A=1/9$, $B=-2/27$, and $C=1/9+4/27=5/27$. Hence, the general solution is
$$y(x)=y_c(x)+y_p(x)=c_1 e^{-3x}+c_2xe^{-3x}+\frac{1}{9}x-\frac{2}{27}x+\frac{5}{27}=\boxed{c _ { 1 } e ^ { - 3 x } + c _ { 2 } x e ^ { - 3 x } + \frac { 1 } { 9 } x + \frac { 4 } { 27 }}.$$

Solve the Differential Equation Using the Method of Undetermined Coefficients ytt+6yt+9y=2+xy ^ { tt } + 6 y ^ { t } + 9 y = 2 + xytt+6yt+9y=2+x

Solve the Differential Equation Using the Method of Undetermined Coefficients $y ^ { tt } + 6 y ^ { t } + 9 y = 2 + x$