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

Options

A. `y=(1)/(2) log |2+x^(2)| +c`
B. `y= (1)/(2) log |1 +x^(2)| +c`
C. `y= log |sqrt(1+x^(2))|+c`
D. None of these

Answer

To solve the differential equation \(\frac{dy}{dx} = \frac{x}{1 + x^2}\), we will follow these steps:

### Step 1: Rewrite the Differential Equation
We start with the given differential equation:
\[ \frac{dy}{dx} = \frac{x}{1 + x^2} \]

### Step 2: Multiply Both Sides by \(dx\)
Next, we multiply both sides by \(dx\) to separate the variables:
\[ dy = \frac{x}{1 + x^2} \, dx \]

### Step 3: Integrate Both Sides
Now, we integrate both sides. The left side integrates to \(y\), and we need to integrate the right side:
\[ \int dy = \int \frac{x}{1 + x^2} \, dx \]

### Step 4: Solve the Right Side Integral
To solve the integral \(\int \frac{x}{1 + x^2} \, dx\), we can use a substitution. Let:
\[ u = 1 + x^2 \quad \Rightarrow \quad du = 2x \, dx \quad \Rightarrow \quad \frac{du}{2} = x \, dx \]
Now, substitute \(u\) into the integral:
\[ \int \frac{x}{1 + x^2} \, dx = \int \frac{1}{u} \cdot \frac{du}{2} = \frac{1}{2} \int \frac{1}{u} \, du \]
The integral of \(\frac{1}{u}\) is \(\ln|u|\):
\[ \frac{1}{2} \ln|u| + C = \frac{1}{2} \ln|1 + x^2| + C \]

### Step 5: Combine Results
Now, we can write the result of the integration:
\[ y = \frac{1}{2} \ln|1 + x^2| + C \]

### Step 6: Final Expression
Thus, the solution to the differential equation is:
\[ y = \frac{1}{2} \ln(1 + x^2) + C \]