To solve the differential equation $y'' - 7y' + 12y = x^5 e^{4x}$, we need to:

1. Find the complementary solution, $y_c$, for the associated homogeneous equation.
2. Identify the form of the particular solution, $y_p$, for the non-homogeneous part, without solving for constants.

Step 1: Complementary Solution

The associated homogeneous equation is: $y'' - 7y' + 12y = 0$ To find $y_c$, we solve the characteristic equation: $r^2 - 7r + 12 = 0$ This factors as: $(r - 3)(r - 4) = 0$ giving roots $r = 3$ and $r = 4$. Therefore, the complementary solution is: $y_c = C_1 e^{3x} + C_2 e^{4x}$ where $C_1$ and $C_2$ are constants.

Step 2: Form of the Particular Solution

Since the right-hand side of the equation is $x^5 e^{4x}$, we apply the method of undetermined coefficients. The exponential term $e^{4x}$ suggests we try a particular solution of the form: $y_p = x^m (A_5 x^5 + A_4 x^4 + A_3 x^3 + A_2 x^2 + A_1 x + A_0)e^{4x}$ However, $e^{4x}$ is already part of the complementary solution, so we increase the multiplicity by multiplying by $x^2$.

Thus, the form of the particular solution is: $y_p = x^2 (A_5 x^5 + A_4 x^4 + A_3 x^3 + A_2 x^2 + A_1 x + A_0)e^{4x}$ where $A_0, A_1, A_2, A_3, A_4,$ and $A_5$ are constants to be determined.

In summary:

• Complementary solution: $y_c = C_1 e^{3x} + C_2 e^{4x}$
• Form of particular solution: $y_p = x^2 (A_5 x^5 + A_4 x^4 + A_3 x^3 + A_2 x^2 + A_1 x + A_0)e^{4x}$

Would you like more details on how to determine $m$ and why we chose this form for $y_p$?

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Related Questions

1. How would we solve for the constants $A_0, A_1, \ldots, A_5$ in the particular solution?
2. What if the characteristic roots had included $r = 4$ with multiplicity 2?
3. How does the choice of form for $y_p$ change if the right-hand side is $x^3 e^{4x}$ instead of $x^5 e^{4x}$?
4. Can we apply the method of undetermined coefficients if the right-hand side includes $\sin(4x)$?
5. How would the complementary solution change if the characteristic polynomial had complex roots?

Tip

When choosing a form for $y_p$, always check if terms overlap with the complementary solution. If they do, increase the multiplicity by multiplying by $x$ to ensure linear independence.