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The given Bernoulli equation is:

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

A Bernoulli equation is of the form:

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

In this case:
- \( P(x) = 1 \)
- \( Q(x) = e^x \)
- \( n = -2 \)

To solve it, we first perform the substitution \( v = y^{1-n} = y^3 \), which will transform the equation into a linear form. The steps are as follows:

1. Differentiate \( v = y^3 \) to get \( \frac{dv}{dx} = 3y^2 \frac{dy}{dx} \).

2. Substitute this into the original equation and simplify.

Let me go ahead and solve it step by step.

### Step 1: Substitution
We substitute \( v = y^3 \), so:

\[
\frac{dv}{dx} = 3y^2 \frac{dy}{dx}
\]

Thus, \( \frac{dy}{dx} = \frac{1}{3y^2} \frac{dv}{dx} \).

### Step 2: Substitute in the original equation
Substitute this into the original equation:

\[
\frac{1}{3y^2} \frac{dv}{dx} + y = e^x y^{-2}
\]

Since \( y^2 = v^{2/3} \), this simplifies into a linear equation in \( v \).

Let me calculate the next steps.The general solution to the Bernoulli equation is given by:

\[
y(x) = \frac{2^{1/3}}{2} \left( C_1 e^{-3x} + 3 e^x \right)^{1/3}
\]

Alternatively, the solution can also be expressed in two other complex forms:

\[
y(x) = \frac{2^{1/3} (-1 - \sqrt{3}i)}{4} \left( C_1 e^{-3x} + 3 e^x \right)^{1/3}
\]
\[
y(x) = \frac{2^{1/3} (-1 + \sqrt{3}i)}{4} \left( C_1 e^{-3x} + 3 e^x \right)^{1/3}
\]

However, the first solution is the real-valued one that is usually preferred in most contexts.

Would you like a more detailed breakdown of the steps leading to this solution, or do you have any other questions?

### Related Questions:
1. How is the Bernoulli equation different from linear differential equations?
2. What are the conditions under which we choose the real versus complex solutions?
3. Can Bernoulli equations always be transformed into linear equations?
4. How does the substitution method work in solving nonlinear equations?
5. What are some common applications of Bernoulli equations in real-world scenarios?

**Tip:** The Bernoulli equation can be solved efficiently by recognizing its special form and using substitutions to convert it into a linear equation.

Solve the Bernoulli equation: dy/dx + y = e^x * y^(-2)

This page provides a detailed step-by-step solution to the Bernoulli equation dy/dx + y = e^x * y^(-2), involving key concepts such as substitution and transformation to linear form. Suitable for undergraduate students studying differential equations.

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