91Ó°ÊÓ

When solving a differential equation, the first step is often to use the method of variable separation. This involves manipulating the equation so that all terms involving the variable y are on one side, and all terms involving the variable x are on the other side. For the given exercise, we start with:
\[\frac{dy}{dx} = \frac{y^2 + 1}{x + 2}\]
We aim to separate y and x. So, we rewrite it as:
\[\frac{dy}{y^2 + 1} = \frac{dx}{x + 2}\]
This separation allows us to deal with each variable independently, facilitating the integration process.

Once we've separated the variables in the differential equation, the next step is to integrate both sides. Each side is integrated with respect to its own variable.
For the left-hand side (LHS):
\[\frac{dy}{y^2 + 1}\]
The integral of this term is \[\tan^{-1} y\].
For the right-hand side (RHS):
\[\frac{dx}{x + 2}\]
The integral of this term is \[\text{ln}|x + 2|\].
Collectively, after integrating both sides, we obtain:
\[\tan^{-1} y = \text{ln}|x + 2| + C \]
Here, C is the constant of integration. Integrating separate parts allows us to analyze and solve the equation more effectively.

Sometimes, after solving a differential equation, we might not get an explicit solution where y is isolated. In such cases, we are left with an implicit solution.
An implicit solution for our example is: \[\tan^{-1} y = \text{ln}|x + 2| + C \]
If it's necessary to solve for y to get an explicit solution, you can manipulate the equation accordingly. For the exercise, we solve for y as follows:
\[\tan^{-1} y = \text{ln}|x + 2| + C\]
Taking the tangent of both sides gives us:
\[\tan (\text{ln}|x + 2| + C) = y\]
However, leaving the solution in its implicit form is perfectly acceptable in many cases. The implicit solution often provides valuable insight into the behavior and constraints of the equation without further manipulation.