The given differential equation is:

\[
(1 - 3t) \frac{dy}{dt} - y = 0
\]

and the initial condition is \( y(2) = -6 \).

This is a **first-order linear differential equation** and can be solved using an integrating factor.

### Step 1: Rewrite the equation in standard form
The standard form of a first-order linear equation is:

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

Rewriting the given equation:
\[
\frac{dy}{dt} - \frac{y}{1 - 3t} = 0
\]
\[
\frac{dy}{dt} + \left(- \frac{1}{1 - 3t}\right)y = 0
\]
Here, \( P(t) = -\frac{1}{1 - 3t} \).

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

\[
\mu(t) = e^{\int P(t) dt}
\]

Substitute \( P(t) = -\frac{1}{1 - 3t} \):

\[
\mu(t) = e^{\int -\frac{1}{1 - 3t} dt}
\]
Let \( u = 1 - 3t \), so \( du = -3 dt \), and we get:

\[
\mu(t) = e^{\frac{1}{3} \ln |1 - 3t|}
\]
\[
\mu(t) = |1 - 3t|^{1/3}
\]

### Step 3: Multiply the original equation by the integrating factor
Now multiply both sides of the equation by \( \mu(t) = |1 - 3t|^{1/3} \):

\[
|1 - 3t|^{1/3} \frac{dy}{dt} + |1 - 3t|^{1/3} \left(- \frac{1}{1 - 3t}\right) y = 0
\]

The left-hand side becomes the derivative of \( y(t) |1 - 3t|^{1/3} \), so the equation simplifies to:

\[
\frac{d}{dt} \left( y(t) |1 - 3t|^{1/3} \right) = 0
\]

### Step 4: Integrate both sides
Integrating both sides gives:

\[
y(t) |1 - 3t|^{1/3} = C
\]
where \( C \) is a constant of integration.

### Step 5: Solve for \( y(t) \)
Solving for \( y(t) \):

\[
y(t) = \frac{C}{|1 - 3t|^{1/3}}
\]

### Step 6: Apply the initial condition \( y(2) = -6 \)
Substitute \( t = 2 \) and \( y(2) = -6 \):

\[
-6 = \frac{C}{|1 - 3(2)|^{1/3}} = \frac{C}{|-5|^{1/3}} = \frac{C}{5^{1/3}}
\]

So:

\[
C = -6 \times 5^{1/3}
\]

### Final solution:
The solution to the differential equation is:

\[
y(t) = \frac{-6 \times 5^{1/3}}{|1 - 3t|^{1/3}}
\]

Would you like more details on any step, or have any questions?

Here are five related questions you can explore:
1. How do you find the integrating factor for other first-order linear differential equations?
2. What happens if the initial condition is given at a different point?
3. How can this method be adapted to non-homogeneous differential equations?
4. What is the significance of the absolute value in the solution?
5. How does the solution behave as \( t \) approaches 1/3?

**Tip:** When solving first-order linear differential equations, always check that your integrating factor is correctly computed; small mistakes here can lead to errors in the solution.

Discover how to solve the differential equation (1 - 3t)y' - y = 0 using the integrating factor method. This problem includes detailed steps and the application of the initial condition y(2) = -6. Explore key mathematical concepts related to first-order linear differential equations and integrating factors.

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