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The problem on the board and the notebook appears to involve solving a differential equation. Specifically, the equation written on the second image is:

\[
\text{tan}(y) + (x + 1)y' = 0
\]

This is a first-order ordinary differential equation (ODE) where \(y' = \frac{dy}{dx}\). To solve it, we can try to isolate \(y'\) and look for a separable or integrable form.

### Steps to solve:
1. **Rearrange the equation:**

\[
(x + 1) y' = -\tan(y)
\]

2. **Separate variables:**
   - Move terms involving \(y\) to one side and terms involving \(x\) to the other:

\[
\frac{dy}{\tan(y)} = -\frac{dx}{x + 1}
\]

3. **Integrate both sides:**

\[
\int \frac{dy}{\tan(y)} = -\int \frac{dx}{x + 1}
\]

Using standard integrals:
- The integral of \(\frac{1}{\tan(y)}\) is \(\ln|\sin(y)|\),
- The integral of \(\frac{1}{x + 1}\) is \(\ln|x + 1|\),

Thus, we have:

\[
\ln|\sin(y)| = -\ln|x + 1| + C
\]

4. **Simplify the solution:**

\[
|\sin(y)| = \frac{A}{|x + 1|}
\]

where \(A = e^C\) is a constant of integration.

This is the general solution to the given differential equation.

Would you like me to provide further details on this solution?

### Follow-up Questions:
1. What are the next steps to solve similar first-order differential equations?
2. How do we interpret constants of integration in different contexts?
3. Can this equation be solved using numerical methods, and if so, how?
4. How would the solution change if the equation was non-homogeneous?
5. How can we verify the solution through substitution?

### Tip:
Always check if the differential equation is separable or exact before starting the solution, as it often simplifies the problem significantly.

Find the general solution of the differential equation: tan(y) + (x + 1)y' = 0

Learn how to solve the first-order differential equation tan(y) + (x + 1)y' = 0 using separation of variables. This problem involves integrating trigonometric and algebraic functions, leading to a general solution for y in terms of x.

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