Solve : `(dy)/(dx)= (1)/(1+x^(2)), y(0)=3`.

Answer

To solve the differential equation \(\frac{dy}{dx} = \frac{1}{1+x^2}\) with the initial condition \(y(0) = 3\), we will follow these steps:

### Step 1: Separate the variables
We can rewrite the equation in a form that allows us to separate the variables:
\[ dy = \frac{1}{1+x^2} \, dx \]

### Step 2: Integrate both sides
Now, we will integrate both sides:
\[ \int dy = \int \frac{1}{1+x^2} \, dx \]
The left side integrates to \(y\), and the right side integrates to \(\tan^{-1}(x) + C\), where \(C\) is the constant of integration:
\[ y = \tan^{-1}(x) + C \]

### Step 3: Apply the initial condition
We know from the problem that \(y(0) = 3\). We will use this to find the constant \(C\):
\[ 3 = \tan^{-1}(0) + C \]
Since \(\tan^{-1}(0) = 0\), we have:
\[ 3 = 0 + C \implies C = 3 \]

### Step 4: Write the particular solution
Now that we have found \(C\), we can write the particular solution:
\[ y = \tan^{-1}(x) + 3 \]

### Final Solution
Thus, the solution to the differential equation with the given initial condition is:
\[ y = \tan^{-1}(x) + 3 \]

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