The equation shown in the image is:

$\frac{dy}{dx} = -3 - e^x y^2$

This is a Bernoulli differential equation, which has the form:

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

Here, the equation can be rearranged into a form that fits the Bernoulli equation structure. To solve it, we proceed as follows:

Step 1: Rewrite the equation

First, we can rewrite the given equation as:

$\frac{dy}{dx} + e^x y^2 = -3$

Here, $P(x) = 0$ and $Q(x) = e^x$, with $n = 2$.

Step 2: Substitute $v = y^{1-n} = y^{-1}$

Let's define a new variable $v = y^{-1}$, so $y = v^{-1}$ and $\frac{dy}{dx} = -v^{-2} \frac{dv}{dx}$.

Step 3: Substitution into the equation

Substitute into the original equation:

$-\frac{1}{v^2} \frac{dv}{dx} + e^x \frac{1}{v^2} = -3$

Multiply through by $-v^2$ to simplify:

$\frac{dv}{dx} = 3v^2 - e^x$

Step 4: Solve the new linear equation

This equation is now linear in $v$. The general approach to solve it is by using an integrating factor or a method appropriate for linear first-order differential equations.

However, let's move forward assuming you want an explicit integration process.

Step 5: Find the integrating factor

For an equation of the form:

$\frac{dv}{dx} + P(x)v = Q(x)$

we need to find the integrating factor $\mu(x) = e^{\int P(x) dx}$. In this case, $P(x) = -3$.

So:

$\mu(x) = e^{-3x}$

Step 6: Multiply the equation by the integrating factor

Multiply the equation by the integrating factor $e^{-3x}$:

$e^{-3x} \frac{dv}{dx} - 3e^{-3x} v = -e^{-2x}$

This simplifies to:

$\frac{d}{dx}(v e^{-3x}) = -e^{-2x}$

Step 7: Integrate both sides

Integrate both sides with respect to $x$:

$v e^{-3x} = \int -e^{-2x} dx$

The integral on the right-hand side is:

$\int -e^{-2x} dx = \frac{1}{2}e^{-2x} + C$

So:

$v e^{-3x} = \frac{1}{2}e^{-2x} + C$

Step 8: Solve for $v$

Solve for $v$:

$v = \frac{1}{2}e^{x} + Ce^{3x}$

Step 9: Substitute back $v = y^{-1}$

Finally, substitute $v = y^{-1}$:

$y = \frac{1}{\frac{1}{2}e^{x} + Ce^{3x}}$

Thus, the solution to the differential equation is:

$y(x) = \frac{2e^{-x}}{1 + 2Ce^{2x}}$

Would you like more details or explanations on any of the steps?

Here are some related questions you might find interesting:

1. What is the general form of a Bernoulli differential equation?
2. How do you derive the integrating factor for a first-order linear differential equation?
3. Can all Bernoulli equations be transformed into linear differential equations?
4. What are the conditions for a differential equation to be exact?
5. How do we solve a first-order separable differential equation?

Tip: Always verify your final solution by substituting it back into the original differential equation to ensure it satisfies the equation.