The equation you've uploaded is:

$(1 + y^2) \, dx = \left( \tan^{-1}(y) - x \right) \, dy$

This appears to be a first-order differential equation. To solve this, we will look for a method such as separation of variables or integrating factor, or we might check if it's an exact equation.

Let's begin solving it step by step:

1. Rearrange the terms: $(1 + y^2) \, dx + (x - \tan^{-1}(y)) \, dy = 0$ This is now in the form of a differential equation $M(x, y) \, dx + N(x, y) \, dy = 0$.

2. Check for exactness: For the equation to be exact, we need to check if the following condition holds: $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$ where:

   • $M(x, y) = 1 + y^2$
   • $N(x, y) = x - \tan^{-1}(y)$

Let's check this condition first to determine the next step.The partial derivatives are:

• $\frac{\partial M}{\partial y} = 2y$
• $\frac{\partial N}{\partial x} = 1$

Since $2y \neq 1$, the equation is not exact. We'll need to explore other methods, such as finding an integrating factor, or potentially using a substitution method to simplify it.

Would you like me to try a specific method to solve this, or would you prefer a more general strategy?

Also, here are five related questions that might deepen your understanding:

1. What is an exact differential equation, and how do we check for it?
2. How do we solve differential equations using integrating factors?
3. What is a separable differential equation, and when is it applicable?
4. How do we handle non-exact differential equations?
5. Can substitution methods always simplify non-exact differential equations?

Tip: Checking for exactness is a useful first step when dealing with first-order differential equations, as it can simplify the solution process significantly.