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The differential equation provided is:

\[
8y' + y = 9t^2
\]

To solve this, we will use the method for solving first-order linear differential equations.

### Step 1: Rewrite the equation

We can rewrite the equation in standard form:

\[
y' + \frac{1}{8}y = \frac{9}{8}t^2
\]

### Step 2: Find the integrating factor

The integrating factor, \( \mu(t) \), is given by:

\[
\mu(t) = e^{\int \frac{1}{8} \, dt} = e^{\frac{t}{8}}
\]

### Step 3: Multiply through by the integrating factor

Multiplying both sides of the equation by \( e^{\frac{t}{8}} \):

\[
e^{\frac{t}{8}}y' + \frac{1}{8}e^{\frac{t}{8}}y = \frac{9}{8}t^2e^{\frac{t}{8}}
\]

The left side is now the derivative of \( y \cdot e^{\frac{t}{8}} \), so we can rewrite the equation as:

\[
\frac{d}{dt} \left( y \cdot e^{\frac{t}{8}} \right) = \frac{9}{8}t^2e^{\frac{t}{8}}
\]

### Step 4: Integrate both sides

We now integrate both sides:

\[
y \cdot e^{\frac{t}{8}} = \int \frac{9}{8}t^2e^{\frac{t}{8}} \, dt
\]

We can solve the right-hand side using integration by parts. The result of the integration is:

\[
y \cdot e^{\frac{t}{8}} = 9 \left( t^2 - 16(t - 8) \right) + C
\]

where \( C \) is the constant of integration. Dividing both sides by \( e^{\frac{t}{8}} \), we get:

\[
y = 9 \left( t^2 - 16(t - 8) \right) + \frac{C}{e^{\frac{t}{8}}}
\]

### Step 5: Behavior as \( t \to \infty \)

As \( t \to \infty \), the term involving \( C/e^{\frac{t}{8}} \) approaches zero because the exponential in the denominator grows very quickly. Therefore, the solution converges to:

\[
y = 9 \left( t^2 - 16(t - 8) \right)
\]

Thus, the function to which the solution converges is:

\[
y = 9 \left( t^2 - 16(t - 8) \right)
\]

Let me know if you'd like more details or clarifications on any step!

### Follow-up Questions:
1. What is the purpose of the integrating factor in solving first-order differential equations?
2. How do we know when to use the method of integrating factors?
3. What other methods could be used to solve this type of differential equation?
4. How does the solution change if the differential equation includes additional terms (e.g., \( y \) terms)?
5. How can we interpret the physical meaning of the solution for large values of \( t \)?

### Tip:
When solving differential equations, always check if the equation is separable, linear, or exact, as the solution method depends on the structure of the equation.

Find the general solution of the given differential equation, and use it to determine how solutions behave as t → ∞.

8y' + y = 9t^2

Learn how to solve the first-order linear differential equation 8y' + y = 9t^2 using the integrating factor method. Explore the step-by-step solution and discover how the solution behaves as t approaches infinity.

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