Let's solve the integral explicitly to find the general solution. 

We were at the step:
\[
e^{t(1 - \ln(t))} y = \int e^{t(1 - \ln(t))} \frac{2}{t^2} \, dt + C
\]

To solve this, simplify the exponent first:
\[
e^{t(1 - \ln(t))} = e^t \cdot e^{-\ln(t) \cdot t} = e^t \cdot t^{-t}
\]
Thus, the integral becomes:
\[
\int e^{t} \cdot t^{-t} \cdot \frac{2}{t^2} \, dt = 2 \int e^t \cdot t^{-t-2} \, dt
\]

This integral is quite complex and does not have a straightforward elementary solution. However, if we focus on a specific context, we might use approximation techniques or special functions to evaluate it. 

To explore a possible approach, let's use a substitution. Define:
\[
u = t
\]
Thus:
\[
du = dt
\]

The integral simplifies to:
\[
2 \int e^{u} \cdot u^{-u-2} \, du
\]

This integral is still complex and likely involves the **Gamma function** or special forms, but since we're working toward a general solution, we'll denote this integral as \( I(t) \) for clarity:

\[
I(t) = \int e^{t} \cdot t^{-t-2} \, dt
\]

So the general solution of the differential equation is:
\[
y(t) = \frac{1}{e^{t(1 - \ln(t))}} \left( 2 I(t) + C \right)
\]

Substituting back:
\[
y(t) = \frac{1}{e^t \cdot t^{-t}} \left( 2 I(t) + C \right)
\]
\[
y(t) = \frac{t^t}{e^t} \left( 2 I(t) + C \right)
\]

### Summary of the General Solution:
\[
y(t) = \frac{t^t}{e^t} \left( 2 \int e^t \cdot t^{-t-2} \, dt + C \right)
\]

This is the general form of the solution, although the integral \( I(t) \) itself is complex and would typically be evaluated using numerical methods or approximated in practice.

Would you like more insights into solving similar integrals or understanding the role of special functions here?

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**Relative Questions:**

1. What are special functions, and how are they used in solving integrals?
2. How does the Gamma function relate to complex integrals?
3. Can you explain the method of substitution in integral calculus?
4. What are some common approaches for approximating difficult integrals?
5. How can numerical integration methods help in solving complex problems?

**Tip:** When encountering difficult integrals, consider whether the problem can be simplified or if the result can be expressed in terms of special functions.

Explore the explicit solution of an integral involving exponential and power functions. Learn how to solve complex integrals step-by-step, including the application of special functions. Suitable for advanced college-level students and professionals.

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