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The given differential equation is:

\[
y' = 9xe^{x^2}
\]

To solve this, we can integrate both sides with respect to \(x\). 

### Step 1: Rewrite the equation

We start by recognizing that \(y' = \frac{dy}{dx}\), so the equation becomes:

\[
\frac{dy}{dx} = 9xe^{x^2}
\]

### Step 2: Integrate both sides

We now integrate the right-hand side with respect to \(x\):

\[
y = \int 9xe^{x^2} \, dx
\]

### Step 3: Solve the integral

The integral can be simplified using the substitution method. Let:

\[
u = x^2 \quad \text{so} \quad du = 2x \, dx
\]

Thus, \(xe^{x^2} \, dx\) becomes \(\frac{1}{2} e^u \, du\). Applying this substitution:

\[
y = \int 9x e^{x^2} \, dx = 9 \cdot \frac{1}{2} \int e^u \, du = \frac{9}{2} e^{x^2}
\]

### Step 4: Add the constant of integration

After integrating, we get:

\[
y = \frac{9}{2} e^{x^2} + C
\]

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

### Final Solution:

The general explicit solution to the differential equation is:

\[
y = \frac{9}{2} e^{x^2} + C
\]

Would you like further details or clarifications on any of the steps?

Here are some related questions you might find helpful:
1. How do you apply substitution in integration problems?
2. Can all first-order differential equations be solved with integration?
3. What is the role of the constant of integration in general solutions?
4. How would you handle this problem if it were a second-order differential equation?
5. What other methods can be used to solve first-order differential equations?

**Tip**: When solving differential equations, always remember to include the constant of integration unless you're solving an initial value problem.

Find the general explicit solution of the differential equation y' = 9xe^(x^2).

Learn how to solve the differential equation y' = 9xe^(x^2) using integration and the substitution method. This step-by-step guide walks you through finding the general explicit solution, suitable for college-level calculus students.

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