To solve the differential equation:

$\frac{dy}{dt} = \frac{9y + 8}{t}$

This is a first-order linear differential equation. We can proceed using the method of separation of variables. Here's the step-by-step solution:

Step 1: Rewrite the equation

We want to separate the variables $y$ and $t$ to be on different sides of the equation. First, write the equation as:

$\frac{dy}{dt} = \frac{9y}{t} + \frac{8}{t}$

Now, rewrite it in the form of $\frac{dy}{dt} = f(t) + g(y)$:

$\frac{dy}{dt} - \frac{9y}{t} = \frac{8}{t}$

This is a linear first-order equation of the form $\frac{dy}{dt} + P(t) y = Q(t)$, where $P(t) = -\frac{9}{t}$ and $Q(t) = \frac{8}{t}$.

Step 2: Find the integrating factor

The integrating factor is given by:

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

Substitute $P(t) = -\frac{9}{t}$:

$\mu(t) = e^{\int -\frac{9}{t} \, dt} = e^{-9 \ln t} = t^{-9}$

Step 3: Multiply through by the integrating factor

Multiply the entire differential equation by $\mu(t) = t^{-9}$:

$t^{-9} \frac{dy}{dt} - \frac{9y}{t} t^{-9} = \frac{8}{t} t^{-9}$

Simplifying:

$t^{-9} \frac{dy}{dt} - 9y t^{-10} = 8 t^{-10}$

Now, the left-hand side is a perfect derivative:

$\frac{d}{dt} \left( y t^{-9} \right) = 8 t^{-10}$

Step 4: Integrate both sides

Now, integrate both sides with respect to $t$:

$\int \frac{d}{dt} \left( y t^{-9} \right) \, dt = \int 8 t^{-10} \, dt$

The left side integrates to $y t^{-9}$, and the right side integrates to:

$\int 8 t^{-10} \, dt = \frac{8}{-9} t^{-9} = -\frac{8}{9} t^{-9}$

So we have:

$y t^{-9} = -\frac{8}{9} t^{-9} + C$

Step 5: Solve for $y$

Now, multiply through by $t^9$ to solve for $y$:

$y = -\frac{8}{9} + C t^9$

Thus, the general solution to the differential equation is:

$y(t) = C t^9 - \frac{8}{9}$

where $C$ is an arbitrary constant determined by initial conditions, if provided.

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Would you like more details or have any questions? Here are some related questions you might find interesting:

1. What is the interpretation of the integrating factor in solving differential equations?
2. How does the separation of variables method work in nonlinear differential equations?
3. Can this method be used for equations with other types of terms, such as non-linear ones?
4. What is the significance of finding the general solution to a differential equation?
5. How would you apply initial conditions to determine the constant $C$?

Tip: Always check whether a differential equation is separable or linear first, as it simplifies the solving process significantly.