The solution of the differential equation `(dy)/(dx)+(1+x^2)/x =0` is

Options

A. `y=-1/2 tan^(-1) x+c `
B. `y+log x + x^2/2 =c`
C. `y=1/2 tan^(-1) x+c `
D. `y-log x - x^2/2 =c `

Answer

To solve the differential equation

\[ \frac{dy}{dx} + \frac{1 + x^2}{x} = 0, \]

we will follow these steps:

### Step 1: Rearranging the Equation
We start by isolating \(\frac{dy}{dx}\):

\[ \frac{dy}{dx} = -\frac{1 + x^2}{x}. \]

### Step 2: Integrating Both Sides
Next, we will integrate both sides with respect to \(x\):

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

### Step 3: Simplifying the Right Side
We can simplify the right-hand side:

\[ -\int \left( \frac{1}{x} + x \right) \, dx = -\left( \int \frac{1}{x} \, dx + \int x \, dx \right). \]

### Step 4: Performing the Integrals
Now we perform the integrals:

1. \(\int \frac{1}{x} \, dx = \log |x|\)
2. \(\int x \, dx = \frac{x^2}{2}\)

Putting it all together, we have:

\[ -\left( \log |x| + \frac{x^2}{2} \right) + C, \]

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

### Step 5: Writing the Final Solution
Thus, we can express \(y\) as:

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

Rearranging gives us:

\[ y + \log |x| + \frac{x^2}{2} = C. \]

### Final Answer
The solution of the differential equation is:

\[ y + \log |x| + \frac{x^2}{2} = C. \]

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