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.