The differential equation given is:

$\frac{dy}{dx} - \frac{\tan y}{1 + x} = (1 + x)e^x \sec y$

Step-by-Step Solution:

1. First, let's isolate $\frac{dy}{dx}$:

$\frac{dy}{dx} = \frac{\tan y}{1 + x} + (1 + x)e^x \sec y$

This is now a first-order nonlinear differential equation. Unfortunately, it doesn't seem to have an easy factorization or direct solution through standard methods like separation of variables or an integrating factor.

However, we can try solving this numerically or approach it through more advanced techniques like substitution or transformation if we recognize any specific patterns (like a Riccati equation).

Would you like me to proceed with a numerical solution, or is there a specific method you'd like me to apply?

Related questions:

1. What is a Riccati equation, and how does it relate to this type of differential equation?
2. How can numerical methods, such as Euler's method, be used to solve this differential equation?
3. How can we transform non-linear differential equations into linear ones using substitution?
4. How does the secant function, $\sec y$, complicate this differential equation compared to a simpler trigonometric function?
5. What are some common techniques to simplify equations involving both $\tan y$ and $\sec y$?

Tip:

When dealing with non-linear differential equations, recognizing the form (e.g., Bernoulli, Riccati) can guide you toward effective solution methods.