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Let's solve these two differential equations one by one:

### Problem 1:
\[
\frac{dx}{dy} = \frac{x^2 + 1}{y + 5}
\]
This is a separable differential equation because we can separate the variables \( x \) and \( y \).

#### Step-by-step Solution:
1. Rewrite the equation to separate the variables:
   \[
   \frac{dx}{x^2 + 1} = \frac{dy}{y + 5}
   \]
   
2. Integrate both sides:
   - The left-hand side is \( \int \frac{1}{x^2 + 1} \, dx \), which equals \( \tan^{-1}(x) \).
   - The right-hand side is \( \int \frac{1}{y + 5} \, dy \), which equals \( \ln|y + 5| \).

   So, after integration:
   \[
   \tan^{-1}(x) = \ln|y + 5| + C
   \]
   where \( C \) is the constant of integration.

3. Solve for \( x \) in terms of \( y \) (if needed), but this is the implicit solution.

### Problem 2:
\[
(xy + y^2)dx + x^2dy = 0
\]
This is an exact differential equation because it can be rewritten as:
\[
M(x, y)dx + N(x, y)dy = 0
\]
where \( M(x, y) = xy + y^2 \) and \( N(x, y) = x^2 \).

#### Step-by-step Solution:
1. Check if the equation is exact by verifying if:
   \[
   \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}
   \]
   - \( \frac{\partial M}{\partial y} = x + 2y \)
   - \( \frac{\partial N}{\partial x} = 2x \)
   
   Since \( x + 2y \neq 2x \), the equation is not exact as it stands.

2. We can attempt to find an integrating factor or use other methods, such as solving it as a non-exact equation (which might involve substitutions or other techniques).

Would you like further details on the second problem, or should we explore other methods for solving it?

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### Follow-up Questions:
1. What is a separable differential equation, and how can it be identified?
2. Can all differential equations be solved by separation of variables?
3. How do you know when a differential equation is exact?
4. What is the role of an integrating factor in solving non-exact equations?
5. How can implicit solutions be converted into explicit ones?

**Tip:** Always check if a differential equation is exact before solving, as it can greatly simplify the problem!

1. dx/dy = (x^2 + 1) / (y + 5)
2. (xy + y^2)dx + x^2dy = 0

Learn how to solve differential equations like dx/dy = (x^2 + 1) / (y + 5) and (xy + y^2)dx + x^2dy = 0. This tutorial covers separable and exact equations, including step-by-step integration and verification methods. Suitable for university-level students.

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