You provided the following differential equation:

\[
\frac{dy}{dt} = -\frac{y}{t} + 6e^t
\]

This is a **first-order linear differential equation**. Let's solve it step-by-step.

### Step 1: Write the equation in standard linear form
We want to express the equation in the form:

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

We have:

\[
\frac{dy}{dt} + \frac{1}{t}y = 6e^t
\]

Here, \(P(t) = \frac{1}{t}\) and \(Q(t) = 6e^t\).

### Step 2: Find the integrating factor
The integrating factor \(\mu(t)\) is given by:

\[
\mu(t) = e^{\int P(t) dt} = e^{\int \frac{1}{t} dt} = e^{\ln |t|} = |t|
\]

Thus, the integrating factor is \(\mu(t) = t\).

### Step 3: Multiply both sides of the equation by the integrating factor
Multiplying the entire differential equation by \(t\), we get:

\[
t \frac{dy}{dt} + y = 6te^t
\]

### Step 4: Integrate both sides
Now, observe that the left-hand side is the derivative of \(y \cdot t\):

\[
\frac{d}{dt}(yt) = 6te^t
\]

Integrating both sides with respect to \(t\), we get:

\[
yt = \int 6te^t dt
\]

To solve the integral on the right-hand side, we use **integration by parts**.

### Step 5: Solve the integral
Let \(u = t\) and \(dv = 6e^t dt\). Then, \(du = dt\) and \(v = 6e^t\). Using the integration by parts formula:

\[
\int u \, dv = uv - \int v \, du
\]

we get:

\[
\int 6te^t dt = 6te^t - 6e^t + C
\]

### Step 6: Solve for \(y\)
Thus, the solution is:

\[
yt = 6te^t - 6e^t + C
\]

Finally, divide by \(t\) to isolate \(y\):

\[
y(t) = 6e^t - \frac{6e^t}{t} + \frac{C}{t}
\]

### Final solution:
\[
y(t) = 6e^t \left(1 - \frac{1}{t}\right) + \frac{C}{t}
\]

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

Let me know if you need further details or explanations.

Here are 5 related questions you might explore:
1. How does the integrating factor simplify solving linear differential equations?
2. Can we apply a different method to solve this differential equation?
3. How would the solution change if the right-hand side function was different?
4. What does the solution look like for specific values of \(t\)?
5. Can you solve a second-order differential equation similarly?

**Tip**: When using integration by parts, always carefully choose your \(u\) and \(dv\) to simplify the integral in the next step.

Learn how to solve the first-order linear differential equation dy/dt = -y/t + 6e^t using the integrating factor method. This problem involves key mathematical concepts such as linear equations, integration by parts, and differential equation techniques. Suitable for Grades 11-12 and college students.

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