https://file.aiquickdraw.com/mathbot/file/1726879779758dxin5s1s.jpg

The equation provided is:

\[
\frac{dy}{dx} - y = -\frac{1}{2}xy^3
\]

This is a Bernoulli differential equation of the form:

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

Where in this case, \( n = 3 \), \( P(x) = -1 \), and \( Q(x) = -\frac{1}{2}x \).

To solve this Bernoulli equation, we can follow these steps:

1. **Divide through by \( y^3 \)** to transform it into a linear differential equation. This gives:
   \[
   \frac{1}{y^3} \frac{dy}{dx} - \frac{1}{y^2} = -\frac{1}{2}x
   \]

2. Introduce a substitution: Let \( v = y^{-2} \), so that \( \frac{dv}{dx} = -2y^{-3} \frac{dy}{dx} \).

3. Solve the resulting linear equation for \( v \).

4. Finally, substitute back for \( y \).

Would you like me to walk you through these steps in more detail and solve the equation?

Solve the Bernoulli differential equation dy/dx - y = -(1/2) * x * y^3 with step-by-step instructions. Learn how to use the substitution method and transform the equation into a linear form. Suitable for undergraduate math students.

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