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Find the complementary solution and the form of the particular solution, but DO NOT solve for the constants in the particular solution: y''-7y'+12y=x^5e^4x

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.

Learn how to find the complementary and particular solutions for the differential equation y'' - 7y' + 12y = x^5e^4x. This guide covers solving for the complementary solution using the characteristic equation and identifying the form of the particular solution with undetermined coefficients, without solving for constants.

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