Consider this second-order nonhomogeneous differential equation:

y''-4y=(x^(2)-3)sin(2x)

What is a suitable form of a particular solution y_(p) of the nonhomogeneous differential equation if the

method of undetermined coefficients is to be used? Use capital letters to represent arbitrary real

constants.

To find a suitable form of a particular solution \( y_p \) for the nonhomogeneous differential equation 

\[
y'' - 4y = (x^2 - 3) \sin(2x),
\]

we can use the method of undetermined coefficients. The right-hand side of the equation is \( (x^2 - 3) \sin(2x) \), which is a product of a polynomial \( x^2 - 3 \) and a trigonometric function \( \sin(2x) \).

### Step 1: Identify the general form

For the right-hand side, we note that:

1. The polynomial part is of degree 2.
2. The trigonometric part is \( \sin(2x) \).

### Step 2: Consider the form of the particular solution

For the method of undetermined coefficients, we propose a particular solution of the form:

\[
y_p = (Ax^2 + Bx + C) \sin(2x) + (Dx^2 + Ex + F) \cos(2x),
\]

where \( A, B, C, D, E, \) and \( F \) are constants to be determined.

### Step 3: Avoiding resonance

Next, we must check if any part of this proposed solution overlaps with the homogeneous solution. The characteristic equation for the homogeneous part \( y'' - 4y = 0 \) is 

\[
r^2 - 4 = 0 \quad \Rightarrow \quad r = \pm 2.
\]

The homogeneous solution is \( y_h = C_1 e^{2x} + C_2 e^{-2x} \). Since \( \sin(2x) \) and \( \cos(2x) \) are not part of the homogeneous solution, our proposed form is suitable.

### Conclusion

Thus, a suitable form for the particular solution \( y_p \) using the method of undetermined coefficients is:

\[
y_p = (Ax^2 + Bx + C) \sin(2x) + (Dx^2 + Ex + F) \cos(2x).
\]

### Questions for Further Exploration:

1. How do you derive the coefficients \( A, B, C, D, E, F \) after substituting \( y_p \) back into the differential equation?
2. What is the complementary solution \( y_c \) of the differential equation?
3. Can you explain why we use undetermined coefficients for this type of nonhomogeneous equation?
4. How would the approach change if the right-hand side were \( e^{2x} \sin(2x) \)?
5. What are some alternative methods to solve nonhomogeneous differential equations if the undetermined coefficients method is not suitable?

### Tip:

Always ensure the proposed form for \( y_p \) is not part of the homogeneous solution; otherwise, you need to multiply by \( x \) to adjust for resonance.

Consider this second-order nonhomogeneous differential equation: y''-4y=(x^(2)-3)sin(2x). What is a suitable form of a particular solution y_(p) of the nonhomogeneous differential equation if the method of undetermined coefficients is to be used?

Explore how to determine a suitable form for a particular solution of the differential equation y''-4y=(x^(2)-3)sin(2x) using the method of undetermined coefficients. This problem is ideal for students studying advanced calculus and differential equations, covering key concepts and methods.

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