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The given differential equation is:

\[
\frac{dy}{dx} = 6x(y - 1)^{2/3}
\]

We can solve this by separation of variables. Let's go step-by-step.

### Step 1: Rewrite the equation

Separate the variables \( y \) and \( x \):

\[
\frac{dy}{(y - 1)^{2/3}} = 6x \, dx
\]

### Step 2: Integrate both sides

#### For the left-hand side (in terms of \( y \)):

\[
\int \frac{1}{(y - 1)^{2/3}} \, dy
\]

We use the substitution \( u = y - 1 \), so \( du = dy \):

\[
\int u^{-2/3} \, du = \frac{u^{1/3}}{1/3} = 3u^{1/3} = 3(y - 1)^{1/3}
\]

Thus, the integral on the left side becomes:

\[
3(y - 1)^{1/3}
\]

#### For the right-hand side (in terms of \( x \)):

\[
\int 6x \, dx = 3x^2
\]

### Step 3: Combine the results

Equating both sides:

\[
3(y - 1)^{1/3} = 3x^2 + C
\]

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

### Step 4: Simplify the equation

Divide both sides by 3:

\[
(y - 1)^{1/3} = x^2 + \frac{C}{3}
\]

Let \( C_1 = \frac{C}{3} \), so the equation becomes:

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

### Step 5: Solve for \( y \)

Cube both sides:

\[
y - 1 = (x^2 + C_1)^3
\]

Finally, solving for \( y \):

\[
y = 1 + (x^2 + C_1)^3
\]

Thus, the general solution to the differential equation is:

\[
y = 1 + (x^2 + C_1)^3
\]

This represents all solutions to the given differential equation.

---

Would you like further details or clarifications on any step? Here are some related questions:

1. How would the solution change for different initial conditions?
2. Can you explain how substitution was used in this problem?
3. What is the role of the constant of integration \( C_1 \)?
4. How can we verify that the solution satisfies the original differential equation?
5. What would happen if \( y = 1 \)? Does it lead to a special case?

**Tip:** Always check for implicit solutions, especially when dealing with non-linear differential equations.

Find all solutions of \( \frac{dy}{dx} = 6x(y - 1)^{2/3} \)

Explore the step-by-step solution to the differential equation \( \frac{dy}{dx} = 6x(y - 1)^{2/3} \), involving the separation of variables, integration, and solving for the general solution. Suitable for undergraduate students studying differential equations.

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