To solve the differential equation

\[
2 \frac{dy}{dt} + 16t^2 y = 2t^2,
\]

we can start by simplifying and rearranging the equation.

1. **Divide through by 2** to make the equation easier to handle:

\[
\frac{dy}{dt} + 8t^2 y = t^2.
\]

Now, this is a first-order linear differential equation of the form:

\[
\frac{dy}{dt} + P(t)y = Q(t),
\]

where \( P(t) = 8t^2 \) and \( Q(t) = t^2 \).

2. **Find the integrating factor** \( \mu(t) \), which is given by:

\[
\mu(t) = e^{\int P(t) \, dt} = e^{\int 8t^2 \, dt}.
\]

Compute the integral:

\[
\int 8t^2 \, dt = \frac{8t^3}{3}.
\]

Therefore, the integrating factor is:

\[
\mu(t) = e^{\frac{8t^3}{3}}.
\]

3. **Multiply the entire differential equation** by the integrating factor \( \mu(t) \):

\[
e^{\frac{8t^3}{3}} \frac{dy}{dt} + 8t^2 e^{\frac{8t^3}{3}} y = t^2 e^{\frac{8t^3}{3}}.
\]

Notice that the left-hand side is the derivative of \( y \cdot \mu(t) \):

\[
\frac{d}{dt} \left( y \cdot e^{\frac{8t^3}{3}} \right) = t^2 e^{\frac{8t^3}{3}}.
\]

4. **Integrate both sides** with respect to \( t \):

\[
y \cdot e^{\frac{8t^3}{3}} = \int t^2 e^{\frac{8t^3}{3}} \, dt.
\]

Let \( u = \frac{8t^3}{3} \), then \( \frac{du}{dt} = 8t^2 \), which implies \( dt = \frac{1}{8t^2} \, du \).

Rewriting the integral:

\[
\int t^2 e^{\frac{8t^3}{3}} \, dt = \frac{1}{8} \int e^u \, du = \frac{1}{8} e^u + C = \frac{1}{8} e^{\frac{8t^3}{3}} + C,
\]

where \( C \) is the constant of integration.

5. **Solve for \( y \)**:

\[
y \cdot e^{\frac{8t^3}{3}} = \frac{1}{8} e^{\frac{8t^3}{3}} + C.
\]

Divide both sides by \( e^{\frac{8t^3}{3}} \):

\[
y = \frac{1}{8} + Ce^{-\frac{8t^3}{3}}.
\]

Thus, the general solution to the differential equation is:

\[
y(t) = \frac{1}{8} + Ce^{-\frac{8t^3}{3}},
\]

where \( C \) is an arbitrary constant.

Would you like more details or have any questions?

Here are some related questions:

1. How do you find the integrating factor for different types of differential equations?
2. What are some common methods for solving first-order linear differential equations?
3. How does the integrating factor method compare to other methods, like separation of variables?
4. Can this technique be used for non-linear differential equations?
5. What are real-world applications of solving differential equations like this one?

**Tip:** Always check if a differential equation is linear or non-linear first, as this will guide you towards the appropriate solving method.

Learn how to solve the differential equation 2 dy/dt + 16t^2 y = 2t^2 step-by-step. This problem involves first-order linear differential equations and the integrating factor method. Suitable for college-level students studying differential equations.

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