This is a first-order linear differential equation:

\[
\frac{dy}{dx} + 3y = 3x^3 - 4x^2 - 5x
\]

We can solve it using an **integrating factor**. The general form of a linear differential equation is:

\[
\frac{dy}{dx} + P(x)y = Q(x)
\]

Here, \( P(x) = 3 \) and \( Q(x) = 3x^3 - 4x^2 - 5x \).

### Step 1: Find the integrating factor
The integrating factor, \( \mu(x) \), is given by:

\[
\mu(x) = e^{\int P(x) \, dx} = e^{\int 3 \, dx} = e^{3x}
\]

### Step 2: Multiply the entire equation by the integrating factor
We multiply both sides of the equation by \( e^{3x} \):

\[
e^{3x} \frac{dy}{dx} + 3e^{3x}y = \left( 3x^3 - 4x^2 - 5x \right) e^{3x}
\]

The left-hand side simplifies to:

\[
\frac{d}{dx} \left( y e^{3x} \right) = \left( 3x^3 - 4x^2 - 5x \right) e^{3x}
\]

### Step 3: Integrate both sides
We now integrate both sides with respect to \(x\):

\[
y e^{3x} = \int \left( 3x^3 - 4x^2 - 5x \right) e^{3x} \, dx
\]

This is an integration by parts problem, and to solve it, we can break down the integrals for each term on the right-hand side.

### Step 4: Solve the integral
We'll solve each term separately:

1. For the term \( 3x^3 e^{3x} \),
2. For \( -4x^2 e^{3x} \),
3. For \( -5x e^{3x} \).

The complete solution involves repeated integration by parts. To simplify things, the result of this integration is:

\[
y e^{3x} = \frac{x^3}{e^{3x}} - \frac{4x^2

Learn how to solve the first-order linear differential equation dy/dx + 3y = 3x^3 - 4x^2 - 5x using integrating factors and integration by parts. This step-by-step guide covers advanced calculus techniques and is ideal for undergraduate students studying differential equations.

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