Solve each of the following differential equations
`(dy)/(dx)-y sin^3 x cos^3 x+xye^(x)=0`.

Answer

To solve the differential equation

\[ \frac{dy}{dx} - y \sin^3 x \cos^3 x + xy e^x = 0, \]

we will follow these steps:

### Step 1: Rearranging the Equation
Rearranging the equation gives us:

\[ \frac{dy}{dx} = y \sin^3 x \cos^3 x - xy e^x. \]

### Step 2: Separating Variables
We can factor out \(y\) on the right-hand side:

\[ \frac{dy}{dx} = y \left( \sin^3 x \cos^3 x - x e^x \right). \]

Next, we separate the variables \(y\) and \(x\):

\[ \frac{dy}{y} = \left( \sin^3 x \cos^3 x - x e^x \right) dx. \]

### Step 3: Integrating Both Sides
Now we integrate both sides:

\[ \int \frac{dy}{y} = \int \left( \sin^3 x \cos^3 x - x e^x \right) dx. \]

The left side integrates to:

\[ \ln |y| = \int \left( \sin^3 x \cos^3 x \right) dx - \int x e^x dx. \]

### Step 4: Solving the Integrals
1. **For the first integral** \(\int \sin^3 x \cos^3 x \, dx\):

Using the identity \(\sin^3 x \cos^3 x = \frac{1}{8} \sin^3(2x)\), we can rewrite it as:

\[ \int \sin^3 x \cos^3 x \, dx = \frac{1}{8} \int \sin^3(2x) \, dx. \]

Using the reduction formula or integration by parts, we can evaluate this integral.

2. **For the second integral** \(\int x e^x \, dx\):

Using integration by parts, let \(u = x\) and \(dv = e^x dx\). Then \(du = dx\) and \(v = e^x\).

Applying integration by parts:

\[ \int x e^x \, dx = x e^x - \int e^x \, dx = x e^x - e^x + C. \]

### Step 5: Combining the Results
Combining both results, we have:

\[ \ln |y| = \frac{1}{8} \int \sin^3(2x) \, dx - (x e^x - e^x) + C. \]

### Step 6: Exponentiating to Solve for \(y\)
Exponentiating both sides gives:

\[ y = e^{\frac{1}{8} \int \sin^3(2x) \, dx - (x e^x - e^x) + C}. \]

This simplifies to:

\[ y = e^C \cdot e^{\frac{1}{8} \int \sin^3(2x) \, dx} \cdot e^{-x e^x + e^x}. \]

Let \(K = e^C\), then:

\[ y = K \cdot e^{\frac{1}{8} \int \sin^3(2x) \, dx} \cdot e^{-x e^x + e^x}. \]

### Final Answer
Thus, the solution to the differential equation is:

\[ y = K \cdot e^{\frac{1}{8} \int \sin^3(2x) \, dx} \cdot e^{-x e^x + e^x}. \]