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.