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To prove the general solution to the differential equation, let's start by analyzing the given differential equation:

\[
3xy^2 y' = 36x^4 + y^3.
\]

We want to solve for \( y(x) \). To simplify this, we can try to rearrange the equation to separate variables or identify a suitable substitution.

### Step 1: Rearrange the Equation

Rewrite the differential equation:

\[
3xy^2 \frac{dy}{dx} = 36x^4 + y^3.
\]

Divide both sides by \( y^2 \) to isolate \( \frac{dy}{dx} \):

\[
3x \frac{dy}{dx} = \frac{36x^4}{y^2} + y.
\]

### Step 2: Identify a Suitable Substitution

To simplify, notice that the equation is nonlinear, and finding a substitution can help reduce it to a simpler form. Let us try the substitution \( y = ux \), where \( u \) is a function of \( x \), and hence:

\[
y' = u'x + u.
\]

Substitute \( y = ux \) and \( y' = u'x + u \) into the differential equation:

\[
3x (ux)^2 (u'x + u) = 36x^4 + (ux)^3.
\]

Simplify the terms:

\[
3x^3 u^2 (u'x + u) = 36x^4 + u^3 x^3.
\]

Divide everything by \( x^3 \):

\[
3u^2 (u'x + u) = 36x + u^3.
\]

Expand and rearrange:

\[
3u^2 u'x + 3u^3 = 36x + u^3.
\]

Combine like terms:

\[
3u^2 u'x = 36x.
\]

Divide by \( x \):

\[
3u^2 u' = 36.
\]

### Step 3: Separate Variables and Integrate

Separate variables:

\[
u^2 du = 12 dx.
\]

Integrate both sides:

\[
\frac{u^3}{3} = 12x + C,
\]

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

Solve for \( u \):

\[
u^3 = 36x + 3C.
\]

Take the cube root:

\[
u = \sqrt[3]{36x + 3C}.
\]

Recall that \( y = ux \), so substitute back:

\[
y = x \sqrt[3]{36x + 3C}.
\]

### Step 4: Simplify to Match the Given Solution

Factor out constants:

\[
y(x) = \sqrt[3]{12x^4 + xC}.
\]

This matches the given general solution, proving its correctness.

### Conclusion

The general solution of the differential equation:

\[
y(x) = \sqrt[3]{12x^4 + xC}.
\]

has been confirmed through the appropriate steps.

Find the general solution of the following differential equation: 3xy^2 y' = 36x^4 + y^3.

Explore the step-by-step solution of the nonlinear differential equation 3xy^2 y' = 36x^4 + y^3. The solution uses the substitution method and separation of variables, providing a detailed explanation suitable for university-level students.

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