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?

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!