https://file.aiquickdraw.com/mathbot/file/17294344392777bu8zw1e.jpg

The differential equation given in the image is:

\[
\frac{dy}{dx} = 1 + xy
\]

This is a first-order non-linear differential equation. To solve it, we can attempt a method called **separation of variables** or apply a known method like an integrating factor, but first, let's check if it can be rearranged into a suitable form.

### Step 1: Rewrite the equation
We have:

\[
\frac{dy}{dx} = 1 + xy
\]

Rewriting it:

\[
\frac{dy}{dx} - xy = 1
\]

This is a linear first-order differential equation of the form:

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

where \( P(x) = -x \) and \( Q(x) = 1 \).

### Step 2: Find the Integrating Factor (IF)
The general solution method for linear equations is to multiply through by an integrating factor, \( \mu(x) \), which is given by:

\[
\mu(x) = e^{\int P(x) \, dx} = e^{\int -x \, dx} = e^{-\frac{x^2}{2}}
\]

### Step 3: Multiply the differential equation by the integrating factor
Multiplying the entire equation by \( e^{-\frac{x^2}{2}} \):

\[
e^{-\frac{x^2}{2}} \frac{dy}{dx} - x e^{-\frac{x^2}{2}} y = e^{-\frac{x^2}{2}}
\]

The left side is now the derivative of \( e^{-\frac{x^2}{2}} y \) with respect to \( x \):

\[
\frac{d}{dx} \left( e^{-\frac{x^2}{2}} y \right) = e^{-\frac{x^2}{2}}
\]

### Step 4: Integrate both sides
Integrating both sides with respect to \( x \):

\[
e^{-\frac{x^2}{2}} y = \int e^{-\frac{x^2}{2}} \, dx
\]

The integral on the right does not have a simple elementary solution, so the solution involves an integral. Thus, the general solution is:

\[
y = e^{\frac{x^2}{2}} \left( \int e^{-\frac{x^2}{2}} \, dx + C \right)
\]

where \( C \) is the constant of integration.

This is the implicit general solution for the differential equation. 

Would you like further details on any step, or explanations on how to handle the integral further?

Here are some follow-up questions:
1. Do you want to explore an alternate method of solving?
2. Should we discuss how to approximate the integral?
3. Would you like to see specific examples or applications of this solution?
4. Do you want to know more about integrating factors in general?
5. Should we discuss how to apply initial conditions if given?

**Tip:** Linear differential equations can often be solved with an integrating factor, which simplifies complex terms into easily integrable expressions.

This tutorial solves the first-order nonlinear differential equation dy/dx = 1 + xy using the integrating factor method. It covers the key steps including finding the integrating factor and integrating the resulting expression. Suitable for undergraduate mathematics students.

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