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

Question

Quizplus · Accepted Answer

The complementary solution is $c_1 e^{-3x} + c_2 x e^{-3x}$. The particular solution for the non-homogeneous part $2 + x$ is $\frac{2}{9}x + \frac{4}{27}$. Combining both gives the complete solution.

Solve the Differential Equation Using the Method of Undetermined Coefficients y′′+6y′+9y=2+xy ^ { \prime \prime } + 6 y ^ { \prime } + 9 y = 2 + xy′′+6y′+9y=2+x

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