The given equation is:

\[
\frac{dx}{dy} = 4x^3
\]

This is a separable differential equation, which we can solve by separating the variables \(x\) and \(y\). Let's follow these steps:

1. Rewrite the equation to separate the variables \(x\) and \(y\):
   \[
   \frac{dx}{x^3} = 4 \, dy
   \]

2. Integrate both sides:
   \[
   \int \frac{dx}{x^3} = \int 4 \, dy
   \]

   The left side can be rewritten as:
   \[
   \int x^{-3} \, dx
   \]

   The integral of \(x^{-3}\) is:
   \[
   \frac{x^{-2}}{-2} = -\frac{1}{2x^2}
   \]

   The integral of \(4\) with respect to \(y\) is:
   \[
   4y + C_1
   \]

   So, after integrating both sides, we get:
   \[
   -\frac{1}{2x^2} = 4y + C_1
   \]

3. Solve for \(x\):
   \[
   \frac{1}{x^2} = -8y - 2C_1
   \]

   To simplify, let \(C_2 = -2C_1\), which gives:
   \[
   \frac{1}{x^2} = -8y + C_2
   \]

4. Finally, solve for \(x\):
   \[
   x = \pm \frac{1}{\sqrt{-8y + C_2}}
   \]

This is the general solution to the differential equation.

Would you like more details on a specific step, or do you have any questions?

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Here are 5 related questions you can explore:
1. How do you solve a first-order separable differential equation?
2. What happens if we change the initial condition for this problem?
3. How would the solution differ if the equation was \(\frac{dx}{dy} = 4x^2\)?
4. What is the geometric interpretation of separable differential equations?
5. How do constants of integration affect the general solution?

**Tip**: Always check if a differential equation is separable before applying more advanced techniques—it simplifies the problem greatly!

Explore the detailed solution to the separable differential equation dx/dy = 4x^3. Learn key calculus concepts, including separation of variables and integration techniques. Suitable for Calculus students at the undergraduate level.

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