You've uploaded an image containing several applied mathematics problems. I'll begin solving a few of them and explain each step clearly. Let's start with the first problem:

Problem 1:

$\frac{dy}{dx} = \tan\left(\frac{y}{x}\right) + \frac{y}{x}$

Solution:

We will attempt a substitution to simplify this equation. Let: $v = \frac{y}{x} \quad \text{or} \quad y = vx$

Now, differentiate $y = vx$ with respect to $x$: $\frac{dy}{dx} = v + x \frac{dv}{dx}$

Substitute this into the original equation: $v + x \frac{dv}{dx} = \tan(v) + v$

Simplify: $x \frac{dv}{dx} = \tan(v)$

This is now a separable differential equation. Separate the variables: $\frac{dv}{\tan(v)} = \frac{dx}{x}$

Integrate both sides: $\int \frac{dv}{\tan(v)} = \int \frac{dx}{x}$

Using the identity $\frac{1}{\tan(v)} = \cot(v)$, we have: $\ln|\sin(v)| = \ln|x| + C$

Exponentiate both sides: $|\sin(v)| = C|x|$

Therefore: $\sin\left(\frac{y}{x}\right) = Cx$

This is the implicit solution to the differential equation.

Would you like a detailed explanation of another specific problem, or shall I continue solving them sequentially? Let me know if you want more details!

Here are 5 related questions that extend this solution:

1. How can we identify the best substitution for complex differential equations?
2. What are the common separable forms in differential equations?
3. Why is the chain rule useful in solving these types of problems?
4. How do you handle boundary conditions or initial values with these solutions?
5. What is the role of implicit solutions in applied mathematics?

Tip: When solving first-order differential equations, look for possible substitutions to simplify expressions.