The equation you provided is:

$y' = 1 + 2xy$

This is a first-order non-linear differential equation. To solve it, we will use an appropriate method such as the integrating factor, substitution, or other relevant techniques based on its structure. Here, it resembles a Bernoulli equation after a transformation.

Let's go through the solution step by step.

Step 1: Rewrite the Equation

The given equation is:

$\frac{dy}{dx} = 1 + 2xy$

We can rearrange this as:

$\frac{dy}{dx} - 2xy = 1$

Step 2: Try the Method of an Integrating Factor

This equation looks like a linear differential equation of the form:

$\frac{dy}{dx} + P(x)y = Q(x)$

Where:

• $P(x) = -2x$
• $Q(x) = 1$

The general method for solving such linear differential equations involves finding an integrating factor $\mu(x)$, which is given by:

$\mu(x) = e^{\int P(x)dx}$

So, for this equation:

$\mu(x) = e^{\int -2x \, dx} = e^{-x^2}$

Step 3: Multiply Through by the Integrating Factor

Multiply both sides of the differential equation by $e^{-x^2}$:

$e^{-x^2} \frac{dy}{dx} - 2xe^{-x^2}y = e^{-x^2}$

The left-hand side is now the derivative of $y \cdot e^{-x^2}$:

$\frac{d}{dx}(y \cdot e^{-x^2}) = e^{-x^2}$

Step 4: Integrate Both Sides

Now integrate both sides with respect to $x$:

$y \cdot e^{-x^2} = \int e^{-x^2} dx$

The integral $\int e^{-x^2} dx$ does not have an elementary function, so we denote it as the error function $\text{erf}(x)$. Therefore, we write the general solution as:

$y \cdot e^{-x^2} = C + \text{erf}(x)$

Where $C$ is the constant of integration.

Step 5: Solve for $y$

Finally, solve for $y$:

$y = \left(C + \text{erf}(x)\right) e^{x^2}$

Final Solution:

$y(x) = \left(C + \text{erf}(x)\right) e^{x^2}$

This is the general solution to the given differential equation.

Would you like more details on any part of this process or have any specific questions?

Related Questions:

1. What are the conditions under which the integrating factor method is applicable?
2. How do you compute the error function $\text{erf}(x)$ in practice?
3. What are some examples of differential equations where the solution includes special functions like $\text{erf}(x)$?
4. Can the integrating factor method be used to solve non-linear equations directly?
5. What other methods can solve first-order non-linear differential equations like this one?

Tip: In some cases, even when an equation has no elementary solution, special functions like the error function can be very useful in expressing the solution!