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To solve this, we are given the differential equation:

\[
\frac{dy}{dx} = 3x^2 \cos \left( 3x^3 + \frac{\pi}{2} \right)
\]

and the point \((0, -1)\). The task is to find an expression for \( y \) in terms of \( x \).

### Step 1: Integrate the given differential equation

We need to integrate both sides of the equation with respect to \( x \):

\[
y = \int 3x^2 \cos \left( 3x^3 + \frac{\pi}{2} \right) dx
\]

Let \( u = 3x^3 + \frac{\pi}{2} \), which implies that \( du = 9x^2 dx \), or equivalently, \( \frac{du}{3} = 3x^2 dx \).

Now, substitute into the integral:

\[
y = \int \cos(u) \frac{du}{3}
\]

This simplifies to:

\[
y = \frac{1}{3} \sin(u) + C
\]

Substitute back for \( u \):

\[
y = \frac{1}{3} \sin \left( 3x^3 + \frac{\pi}{2} \right) + C
\]

### Step 2: Apply the initial condition

We know that when \( x = 0 \), \( y = -1 \). Substitute these values into the equation to solve for \( C \):

\[
-1 = \frac{1}{3} \sin \left( 3(0)^3 + \frac{\pi}{2} \right) + C
\]

\[
-1 = \frac{1}{3} \sin \left( \frac{\pi}{2} \right) + C
\]

Since \( \sin \left( \frac{\pi}{2} \right) = 1 \), we get:

\[
-1 = \frac{1}{3} + C
\]

Solving for \( C \):

\[
C = -1 - \frac{1}{3} = -\frac{4}{3}
\]

### Step 3: Write the final solution

Now that we know \( C = -\frac{4}{3} \), the final expression for \( y \) in terms of \( x \) is:

\[
y = \frac{1}{3} \sin \left( 3x^3 + \frac{\pi}{2} \right) - \frac{4}{3}
\]

Would you like further details or have any questions?

Here are 5 related questions for further exploration:
1. How does substitution help in solving integrals like this one?
2. What are the rules for integrating trigonometric functions?
3. How do you find particular solutions to differential equations using initial conditions?
4. What other methods could be used to solve differential equations?
5. How does the choice of substitution simplify complex integrals?

**Tip**: Always verify if the initial condition is satisfied after solving the differential equation to avoid errors.

Given that dy/dx = 3x^2 * cos(3x^3 + π/2) and that the graph of y passes through the point (0, -1), find an expression for y in terms of x.

Learn how to solve the differential equation dy/dx = 3x^2 cos(3x^3 + π/2) by integrating both sides. This problem covers key calculus concepts such as integration with substitution and the use of initial conditions. Suitable for advanced high school or college-level students.

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