Write the general solution of the following differential equations
`(dy)/(dx)=x^5+x^2 - (2)/(x)`

Answer

To find the general solution of the differential equation

\[ \frac{dy}{dx} = x^5 + x^2 - \frac{2}{x}, \]

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

### Step 1: Rewrite the equation
We start with the given equation:

\[ \frac{dy}{dx} = x^5 + x^2 - \frac{2}{x}. \]

### Step 2: Integrate both sides
To find \(y\), we integrate the right-hand side:

\[ y = \int \left( x^5 + x^2 - \frac{2}{x} \right) dx. \]

### Step 3: Break down the integral
We can break this integral into three separate integrals:

\[ y = \int x^5 \, dx + \int x^2 \, dx - 2 \int \frac{1}{x} \, dx. \]

### Step 4: Calculate each integral
Now we calculate each integral one by one:

1. \(\int x^5 \, dx = \frac{x^6}{6}\)
2. \(\int x^2 \, dx = \frac{x^3}{3}\)
3. \(-2 \int \frac{1}{x} \, dx = -2 \ln |x|\)

### Step 5: Combine the results
Putting it all together, we have:

\[ y = \frac{x^6}{6} + \frac{x^3}{3} - 2 \ln |x| + C, \]

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

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

\[ y = \frac{x^6}{6} + \frac{x^3}{3} - 2 \ln |x| + C. \]

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