We are given the first-order linear differential equation:

$t \frac{dy}{dt} + 6y = 3t - 2, \quad t > 0$ with the initial condition:

$y(1) = 0.$

Step 1: Rewrite the equation in standard form

First, divide the entire equation by $t$ to express it in the standard linear form:

$\frac{dy}{dt} + \frac{6}{t}y = 3 - \frac{2}{t}.$

Step 2: Find the integrating factor

The integrating factor (IF) for a first-order linear differential equation of the form $\frac{dy}{dt} + P(t) y = Q(t)$ is given by:

$\mu(t) = e^{\int P(t) dt}.$

Here, $P(t) = \frac{6}{t}$, so we calculate the integrating factor:

$\mu(t) = e^{\int \frac{6}{t} dt} = e^{6 \ln t} = t^6.$

Step 3: Multiply the differential equation by the integrating factor

Now multiply the entire equation by $t^6$:

$t^6 \frac{dy}{dt} + t^6 \cdot \frac{6}{t} y = t^6 \left( 3 - \frac{2}{t} \right).$

Simplifying:

$t^6 \frac{dy}{dt} + 6t^5 y = 3t^6 - 2t^5.$

Step 4: Solve the equation

Notice that the left-hand side is the derivative of $t^6 y$:

$\frac{d}{dt} \left( t^6 y \right) = 3t^6 - 2t^5.$

Now integrate both sides with respect to $t$:

$t^6 y = \int \left( 3t^6 - 2t^5 \right) dt.$

Integrating the right-hand side:

$t^6 y = \frac{3}{7} t^7 - \frac{2}{6} t^6 + C = \frac{3}{7} t^7 - \frac{1}{3} t^6 + C.$

Step 5: Solve for $y(t)$

Now, solve for $y(t)$:

$y(t) = \frac{1}{t^6} \left( \frac{3}{7} t^7 - \frac{1}{3} t^6 + C \right).$

Simplify:

$y(t) = \frac{3}{7} t - \frac{1}{3} + \frac{C}{t^6}.$

Step 6: Apply the initial condition

We are given that $y(1) = 0$. Substitute $t = 1$ and $y(1) = 0$ into the equation:

$0 = \frac{3}{7} \cdot 1 - \frac{1}{3} + \frac{C}{1^6}.$

Simplifying:

$0 = \frac{3}{7} - \frac{1}{3} + C.$

To solve for $C$, find a common denominator between $\frac{3}{7}$ and $\frac{1}{3}$:

$0 = \frac{9}{21} - \frac{7}{21} + C = \frac{2}{21} + C.$

Thus:

$C = -\frac{2}{21}.$

Step 7: Write the final solution

Substitute $C = -\frac{2}{21}$ back into the solution for $y(t)$:

$y(t) = \frac{3}{7} t - \frac{1}{3} - \frac{2}{21 t^6}.$

Thus, the solution to the initial-value problem is:

$y(t) = \frac{3}{7} t - \frac{1}{3} - \frac{2}{21 t^6}.$

Would you like further details or have any questions?

Here are five related questions to consider:

1. How do you find the integrating factor for a linear differential equation?
2. Why is it important to rewrite the differential equation in standard form before solving?
3. How does the initial condition influence the final solution of a differential equation?
4. What role does the integrating factor play in solving linear differential equations?
5. How would the solution change if the initial condition were different?

Tip: Always check that the initial condition satisfies your final solution to ensure correctness.