Solve the differential equation $y ^ {tt} - 6 y ^ { t } + 9 y = 8 x ^ { 2 } e ^ { 3 x } \text { by the method of variation of }$ parameters.
A) $y = \left( C _ { 1 } + C _ { 2 } x + \frac { 2 } { 3 } x ^ { 4 } \right) e ^ { 3 x }$
B) $y = \left( C _ { 1 } + C _ { 2 } x + \frac { 8 } { 15 } x ^ { 4 } \right) e ^ { 3 x }$
C)$y = \left( C _ { 1 } + C _ { 2 } x + \frac { 4 } { 9 } x ^ { 2 } \right) e ^ { 3 x }$
D)$y = \left( C _ { 1 } + C _ { 2 } x - \frac { 8 } { 17 } x ^ { 2 } \right) e ^ { 3 x }$
E)$y = \left( C _ { 1 } + C _ { 2 } x - \frac { 4 } { 5 } x ^ { 2 } \right) e ^ { 3 x }$

Question

Solve the differential equation $y ^ {tt} - 6 y ^ { t } + 9 y = 8 x ^ { 2 } e ^ { 3 x } \text { by the method of variation of }$ parameters. 
A) $y = \left( C _ { 1 } + C _ { 2 } x + \frac { 2 } { 3 } x ^ { 4 } \right) e ^ { 3 x }$
B) $y = \left( C _ { 1 } + C _ { 2 } x + \frac { 8 } { 15 } x ^ { 4 } \right) e ^ { 3 x }$
C)$y = \left( C _ { 1 } + C _ { 2 } x + \frac { 4 } { 9 } x ^ { 2 } \right) e ^ { 3 x }$
D)$y = \left( C _ { 1 } + C _ { 2 } x - \frac { 8 } { 17 } x ^ { 2 } \right) e ^ { 3 x }$
E)$y = \left( C _ { 1 } + C _ { 2 } x - \frac { 4 } { 5 } x ^ { 2 } \right) e ^ { 3 x }$

Quizplus · Accepted Answer

The complementary solution is $y_c = (C_1 + C_2 x) e^{3x}$. Using variation of parameters, the particular solution is found to be $\frac{2}{3} x^4 e^{3x}$. Thus, the general solution is $y = (C_1 + C_2 x + \frac{2}{3} x^4) e^{3x}$.

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