General solution of `(x^(2)+1)(dy)/(dx)=2` is _________ .

Answer

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

### Step 1: Rewrite the equation
We start by rewriting the given equation to isolate \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{2}{x^2 + 1} \]

### Step 2: Integrate both sides
Next, we integrate both sides with respect to \(x\):
\[ \int dy = \int \frac{2}{x^2 + 1} \, dx \]

### Step 3: Solve the left side
The left side integrates to:
\[ y = \int dy = y + C_1 \]
where \(C_1\) is the constant of integration.

### Step 4: Solve the right side
The right side can be integrated using the known integral:
\[ \int \frac{2}{x^2 + 1} \, dx = 2 \tan^{-1}(x) + C_2 \]
where \(C_2\) is another constant of integration.

### Step 5: Combine the results
Combining the results from both sides, we have:
\[ y + C_1 = 2 \tan^{-1}(x) + C_2 \]

### Step 6: Simplify the equation
We can combine the constants \(C_1\) and \(C_2\) into a single constant \(C\):
\[ y = 2 \tan^{-1}(x) + C \]

### Final Answer
Thus, the general solution of the differential equation is:
\[ y = 2 \tan^{-1}(x) + C \]

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