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Let's break down the steps to solve the differential equation \( y'' - 2y' = (5t - 3) \sin(2t) \):

### Part a: Solve the Homogeneous Equation

We start by solving the homogeneous equation:
\[
y'' - 2y' = 0
\]
This simplifies to:
\[
y'' = 2y'
\]
The characteristic equation is:
\[
r^2 - 2r = 0
\]
Factoring:
\[
r(r - 2) = 0
\]
So, the roots are \( r = 0 \) and \( r = 2 \).

Thus, the general solution to the homogeneous equation is:
\[
y_c = c_1 + c_2 e^{2t}
\]

### Part b: Apply the Method of Undetermined Coefficients

Now we solve the non-homogeneous equation:
\[
y'' - 2y' = (5t - 3) \sin(2t)
\]

Since the right-hand side involves a product of a polynomial \( (5t - 3) \) and a sinusoidal function \( \sin(2t) \), we propose a particular solution of the form:
\[
y_p = t(A t + B) \sin(2t) + t(C t + D) \cos(2t)
\]
We use this ansatz because both sine and cosine terms will account for both the trigonometric function and its polynomial coefficient.

Now, we need to plug this into the original differential equation and determine the coefficients \( A \), \( B \), \( C \), and \( D \) by matching terms on both sides. This part involves differentiating \( y_p \) and comparing terms with \( (5t - 3) \sin(2t) \), which is a bit lengthy but follows from solving systems of equations for these coefficients.

Let me know if you'd like the detailed step-by-step solution for the particular solution!

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Do you want more details on the calculation for the particular solution? Here are some related questions you might find useful:

1. How do we differentiate \( y_p \) to substitute into the differential equation?
2. How do we decide the form of the particular solution in the method of undetermined coefficients?
3. What do we do if the trial solution overlaps with the homogeneous solution?
4. How do we handle non-homogeneous equations with different types of right-hand sides (exponential, polynomial, etc.)?
5. How do you solve the system of equations once we substitute the particular solution into the equation?

**Tip:** Always check if your proposed particular solution overlaps with the homogeneous solution, as this might require adjustments to the trial function (usually by multiplying by \( t \)).

Consider the differential equation: y'' - 2y' = (5t - 3)sin(2t).

Learn how to solve the differential equation y'' - 2y' = (5t - 3)sin(2t) by finding the homogeneous solution and applying the method of undetermined coefficients for the particular solution. Suitable for undergraduate calculus students.

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