Find a particular solution to the differential equation y''-2y'-3y=-9t^3

We are tasked with finding a particular solution to the differential equation:

\[
y'' - 2y' - 3y = -9t^3
\]

### Step 1: Solve for the Particular Solution
Since the non-homogeneous term is a polynomial \( -9t^3 \), we will assume that the particular solution \( y_p(t) \) has the form of a cubic polynomial:

\[
y_p(t) = At^3 + Bt^2 + Ct + D
\]

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

### Step 2: Compute the Derivatives
To substitute into the differential equation, we need the first and second derivatives of \( y_p(t) \):

1. \( y_p'(t) = 3At^2 + 2Bt + C \)
2. \( y_p''(t) = 6At + 2B \)

### Step 3: Substitute into the Differential Equation
Now, substitute \( y_p(t) \), \( y_p'(t) \), and \( y_p''(t) \) into the original equation \( y'' - 2y' - 3y = -9t^3 \):

\[
(6At + 2B) - 2(3At^2 + 2Bt + C) - 3(At^3 + Bt^2 + Ct + D) = -9t^3
\]

Simplifying each term:

\[
6At + 2B - (6At^2 + 4Bt + 2C) - (3At^3 + 3Bt^2 + 3Ct + 3D)
\]

Now, group the terms by powers of \( t \):

\[
-3At^3 - (6A + 3B)t^2 + (6A - 4B - 3C)t + (2B - 2C - 3D)
\]

Set this equal to \( -9t^3 + 0t^2 + 0t + 0 \). Now, equate the coefficients of corresponding powers of \( t \) on both sides:

1. Coefficient of \( t^3 \): \( -3A = -9 \) ⟹ \( A = 3 \)
2. Coefficient of \( t^2 \): \( -6A - 3B = 0 \) ⟹ \( -18 - 3B = 0 \) ⟹ \( B = -6 \)
3. Coefficient of \( t \): \( 6A - 4B - 3C = 0 \) ⟹ \( 18 + 24 - 3C = 0 \) ⟹ \( C = 14 \)
4. Constant term: \( 2B - 2C - 3D = 0 \) ⟹ \( -12 - 28 - 3D = 0 \) ⟹ \( D = -40/3 \)

### Step 4: Write the Particular Solution
Now we have the values for \( A \), \( B \), \( C \), and \( D \). Therefore, the particular solution is:

\[
y_p(t) = 3t^3 - 6t^2 + 14t - \frac{40}{3}
\]

This is the particular solution to the given non-homogeneous differential equation.

Would you like to see the general solution or details about this process? Let me know if you have any questions!

Here are 5 related questions:
1. How do you find the general solution to a non-homogeneous differential equation?
2. What method can you use to solve a differential equation with sinusoidal forcing terms?
3. How would the solution differ if the non-homogeneous term was \( e^{t} \) instead of a polynomial?
4. What is the method of undetermined coefficients in solving differential equations?
5. Can you always assume a polynomial form for the particular solution when the forcing term is a polynomial?

**Tip**: When solving non-homogeneous differential equations, always check the nature of the non-homogeneous term (e.g., polynomial, exponential, trigonometric) to guide your choice of a trial solution.

Find a particular solution to the differential equation y'' - 2y' - 3y = -9t^3

Explore how to find the particular solution to the differential equation y'' - 2y' - 3y = -9t^3 using the method of undetermined coefficients. This guide walks through solving for a polynomial forcing term, step by step, suitable for undergraduate calculus students.

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