Given the differential equation:

\[
(y^2 + 1) \, dx = y \sec(x) \, dy
\]

We need to solve this differential equation. Let's follow these steps:

### Step 1: Separate the variables
We want to rearrange the equation so that all terms involving \( y \) are on one side, and all terms involving \( x \) are on the other side.

\[
\frac{y^2 + 1}{y} \, dx = \sec(x) \, dy
\]

Simplify the left-hand side:

\[
\left( y + \frac{1}{y} \right) dx = \sec(x) \, dy
\]

Now separate the variables \( x \) and \( y \):

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

### Step 2: Integrate both sides
Integrate both sides with respect to their respective variables.

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

#### Left-hand side:
\[
\int \cos(x) \, dx = \sin(x) + C_1
\]

#### Right-hand side:
For the right side, we'll rewrite the integral to separate the terms:

\[
\int \frac{dy}{y + \frac{1}{y}} = \int \frac{y \, dy}{y^2 + 1}
\]

Let \( u = y^2 + 1 \), then \( du = 2y \, dy \), so:

\[
\frac{1}{2} \int \frac{du}{u} = \frac{1}{2} \ln|u| + C_2 = \frac{1}{2} \ln(y^2 + 1) + C_2
\]

### Step 3: Combine the solutions
Putting it all together:

\[
\sin(x) + C_1 = \frac{1}{2} \ln(y^2 + 1) + C_2
\]

Combine the constants \( C_1 \) and \( C_2 \) into a single constant \( C \):

\[
\sin(x) = \frac{1}{2} \ln(y^2 + 1) + C
\]

### Final Answer:

\[
\sin(x) - C = \frac{1}{2} \ln(y^2 + 1)
\]

You can leave it in this form or solve for \( y \) explicitly:

\[
\ln(y^2 + 1) = 2(\sin(x) - C)
\]

\[
y^2 + 1 = e^{2(\sin(x) - C)}
\]

\[
y^2 = e^{2(\sin(x) - C)} - 1
\]

\[
y = \pm \sqrt{e^{2(\sin(x) - C)} - 1}
\]

Let me know if you need more details or have any questions.

### Follow-Up Questions:
1. How can this method be applied to other types of separable differential equations?
2. What is the significance of the constant of integration in the solution?
3. How does the choice of \( u = y^2 + 1 \) simplify the integration process?
4. Can this method be used to solve non-separable differential equations?
5. What are the potential issues if the equation had non-elementary integrals?

### Tip:
Always double-check the steps of integration, especially when dealing with logarithmic functions, to avoid errors.

Learn how to solve the differential equation (y^2 + 1) dx = y sec(x) dy step-by-step with a detailed solution. This problem involves concepts of differential equations, separable equations, and integration techniques. Suitable for advanced undergraduate students and anyone interested in differential equations.

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