We are given the differential equation:

$\frac{dy}{dx} = 2x(y + 1)$

To solve this using separation of variables, follow these steps:

Step 1: Rearrange to separate variables

We want to get all terms involving $y$ on one side and all terms involving $x$ on the other side. First, rewrite the equation as:

$\frac{dy}{y+1} = 2x \, dx$

Step 2: Integrate both sides

Now integrate both sides:

$\int \frac{dy}{y+1} = \int 2x \, dx$

Left side:

The integral of $\frac{1}{y+1}$ is $\ln|y+1|$:

$\ln|y+1| = \int \frac{1}{y+1} \, dy$

Right side:

The integral of $2x$ is $x^2$:

$x^2 + C$

So, the equation becomes:

$\ln|y+1| = x^2 + C$

Step 3: Solve for $y$

To solve for $y$, exponentiate both sides to eliminate the logarithm:

$|y+1| = e^{x^2 + C}$

Since $e^{C}$ is a constant, we can replace it with a new constant, say $A$, giving us:

$y + 1 = A e^{x^2}$

Finally, solve for $y$:

$y = A e^{x^2} - 1$

Step 4: General Solution

Thus, the general solution to the differential equation is:

$y = A e^{x^2} - 1$

where $A$ is a constant of integration.

Would you like further details or have any questions?

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Related Questions:

1. How do you determine the constant of integration in a differential equation?
2. What is the method of integrating factors for solving first-order differential equations?
3. How would you solve the differential equation if it had an initial condition?
4. What are the other methods besides separation of variables for solving differential equations?
5. Can separation of variables be used for partial differential equations as well?

Tip: When integrating, always remember to include the constant of integration to account for all possible solutions.