https://file.aiquickdraw.com/mathbot/file/1726544248280fz4qv7jv.jpg

Let's go over each question and work through the solutions one by one.

### Question 7:
The differential equation is:
\[
\frac{dy}{dt} = 8t \sec\left(\frac{y}{t}\right) + y
\]

This appears to be a homogeneous equation, which often involves a substitution like \( v = \frac{y}{t} \) to simplify the equation. Let's check the solution process for it.

### Question 8:
The differential equation is:
\[
\frac{dy}{dx} = \frac{y(\ln(4y) - \ln(4x) + 9)}{x}
\]
The hint given is:
\[
\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)
\]

Using logarithmic properties and solving through substitution can simplify this equation.

### Question 9:
The Bernoulli equation is given by:
\[
\frac{dy}{dx} + \frac{y}{x} = 7x^8 y^2
\]

This is a Bernoulli equation, and it can be solved by transforming it into a linear equation using the substitution \( v = y^{1-n} \) where \( n \) is the exponent of \( y \) on the right-hand side.

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Now, let me solve these equations in detail.It looks like I mistakenly treated the variable \( y \) as a function in the first equation. Let me correct that and proceed with the solution. I'll make sure that the function forms are properly declared before solving.It seems like I can’t do more advanced data analysis right now. Please try again later.

In the meantime, if you would like me to help explain the general approach to solving these differential equations step by step, feel free to ask! I can guide you through the methods.

Here are 5 relative questions that could further enhance your understanding:

1. What are the general methods for solving homogeneous differential equations?
2. How do logarithmic properties simplify differential equations in cases like Question 8?
3. What are the steps involved in solving a Bernoulli equation like in Question 9?
4. How does substitution simplify complex differential equations?
5. How do you determine whether a solution is lost during the process?

**Tip**: When working with Bernoulli equations, transforming them into linear form can simplify solving them significantly.

Solve the homogenous equation and Bernoulli equation provided in the image, ignoring lost solutions, if any.

This tutorial covers the process of solving a homogeneous differential equation and a Bernoulli equation step by step, highlighting substitution methods and key logarithmic properties. Suitable for students studying calculus or differential equations at a university level.

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