Write the general solution of the following differential equations
`(dy)/(dx)=x^3+e^(x)+x^(e )`

Answer

To solve the differential equation

\[ \frac{dy}{dx} = x^3 + e^x + x^e, \]

we will integrate both sides with respect to \(x\).

### Step 1: Integrate both sides
We start by integrating the right-hand side:

\[ \int \frac{dy}{dx} \, dx = \int \left( x^3 + e^x + x^e \right) \, dx. \]

This gives us:

\[ y = \int \left( x^3 + e^x + x^e \right) \, dx. \]

### Step 2: Break down the integral
Now, we can break down the integral into three separate parts:

\[ y = \int x^3 \, dx + \int e^x \, dx + \int x^e \, dx. \]

### Step 3: Solve each integral
1. **Integral of \(x^3\)**:
\[ \int x^3 \, dx = \frac{x^{3+1}}{3+1} = \frac{x^4}{4}. \]

2. **Integral of \(e^x\)**:
\[ \int e^x \, dx = e^x. \]

3. **Integral of \(x^e\)**:
Using the formula for the integral of \(x^n\):
\[ \int x^e \, dx = \frac{x^{e+1}}{e+1}. \]

### Step 4: Combine the results
Now, we can combine all the results:

\[ y = \frac{x^4}{4} + e^x + \frac{x^{e+1}}{e+1} + C, \]

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

### Final Answer
Thus, the general solution of the differential equation is:

\[ y = \frac{x^4}{4} + e^x + \frac{x^{e+1}}{e+1} + C. \]

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