To solve the given differential equation \((y^2 + 1) \, dx = y \sec^2(x) \, dy\), let's proceed by separating the variables.

### Step 1: Rearrange the equation
We can rearrange the terms to separate \(x\) and \(y\) variables:

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

This simplifies to:

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

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

**Left side:**
\[
\int \cos^2(x) \, dx
\]

We can use the identity \(\cos^2(x) = \frac{1 + \cos(2x)}{2}\) to rewrite the integral:

\[
\int \cos^2(x) \, dx = \frac{1}{2} \int (1 + \cos(2x)) \, dx
\]

This gives:

\[
\frac{1}{2} \left( x + \frac{\sin(2x)}{2} \right) + C_1
\]

Or:

\[
\frac{x}{2} + \frac{\sin(2x)}{4} + C_1
\]

**Right side:**
\[
\int \frac{y \, dy}{y^2 + 1}
\]

This is a standard integral:

\[
\int \frac{y \, dy}{y^2 + 1} = \frac{1}{2} \ln(y^2 + 1) + C_2
\]

### Step 3: Combine the results
Now, equate the two integrals:

\[
\frac{x}{2} + \frac{\sin(2x)}{4} + C_1 = \frac{1}{2} \ln(y^2 + 1) + C_2
\]

To simplify, we can combine the constants \(C_1\) and \(C_2\) into a single constant \(C\):

\[
\frac{x}{2} + \frac{\sin(2x)}{4} = \frac{1}{2} \ln(y^2 + 1) + C
\]

### Step 4: Solve for the general solution
The general solution to the differential equation is:

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

or equivalently:

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

where \(C' = 2C\) is the general constant of integration.

This is the implicit solution to the differential equation.

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Would you like to explore this solution in more detail or ask any other questions?

**Related Questions:**
1. How can you solve separable differential equations in general?
2. What are some common techniques for integrating trigonometric identities?
3. How does the integration of \(\frac{1}{y^2 + 1}\) compare with other standard integrals?
4. What methods can be used to solve differential equations that aren't separable?
5. How can initial conditions be applied to find the particular solution of a differential equation?

**Tip:** Always look for opportunities to simplify trigonometric identities before integrating; it often makes the process much easier.

Learn how to solve the differential equation (y^2 + 1)dx = y sec^2(x) dy step-by-step with detailed explanations. This problem involves separable differential equations and trigonometric integrals, suitable for undergraduate students studying differential equations.

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