The given differential equation is:

\[
\frac{dy}{dt} + \frac{4y}{t} = 7t^4
\]

This is a first-order linear differential equation. To solve it, we can use an **integrating factor** method. Let's follow these steps:

1. Write the equation in standard form:
   \[
   \frac{dy}{dt} + P(t) y = Q(t)
   \]
   Here, \( P(t) = \frac{4}{t} \) and \( Q(t) = 7t^4 \).

2. The integrating factor \( \mu(t) \) is found using:
   \[
   \mu(t) = e^{\int P(t) \, dt} = e^{\int \frac{4}{t} \, dt}
   \]
   \[
   \mu(t) = e^{4 \ln|t|} = t^4
   \]

3. Multiply both sides of the original equation by the integrating factor \( t^4 \):
   \[
   t^4 \frac{dy}{dt} + t^4 \cdot \frac{4y}{t} = t^4 \cdot 7t^4
   \]
   Simplifying:
   \[
   t^4 \frac{dy}{dt} + 4t^3 y = 7t^8
   \]
   The left side is the derivative of \( t^4 y \):
   \[
   \frac{d}{dt}(t^4 y) = 7t^8
   \]

4. Integrate both sides:
   \[
   t^4 y = \int 7t^8 \, dt = \frac{7t^9}{9} + C
   \]
   where \( C \) is the constant of integration.

5. Solve for \( y \):
   \[
   y = \frac{7t^9}{9t^4} + \frac{C}{t^4}
   \]
   Simplifying:
   \[
   y(t) = \frac{7t^5}{9} + \frac{C}{t^4}
   \]

Thus, the general solution is:

\[
y(t) = \frac{7t^5}{9} + \frac{C}{t^4}
\]

Do you need further details or have any questions?

Here are 5 related questions to extend your understanding:
1. What is an integrating factor and how is it useful in solving linear differential equations?
2. How do we recognize when to use the integrating factor method?
3. Could this equation be solved using any other method, like variation of parameters?
4. How would the solution change if the equation had an initial condition, such as \( y(1) = 2 \)?
5. Can you solve a similar equation with different coefficients, for example, \( \frac{dy}{dt} + \frac{5y}{t} = 3t^4 \)?

**Tip:** When solving differential equations, always check if the equation can be written in a recognizable form, like linear or separable, to choose the right solving technique.

Learn how to solve the first-order linear differential equation dy/dt + (4y)/t = 7t^4 using the integrating factor method. This detailed step-by-step solution covers key concepts in differential equations, including integrating factors and solving for the general solution. Suitable for students in Grades 11-12 or college level.

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