Find the general solution of the following differential equations :
`(x^(2)+1)(dy)/(dx)=xy`.

Answer

To find the general solution of the differential equation

\[ (x^2 + 1) \frac{dy}{dx} = xy, \]

we will follow these steps:

### Step 1: Separate the Variables
We start by rearranging the equation to separate the variables \(y\) and \(x\):

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

### Step 2: Integrate Both Sides
Next, we integrate both sides:

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

The left side integrates to:

\[ \log |y| + C_1, \]

where \(C_1\) is a constant of integration.

For the right side, we can use substitution. Let \(u = x^2 + 1\), then \(du = 2x \, dx\) or \(\frac{du}{2} = x \, dx\). Thus, we have:

\[ \int \frac{x}{x^2 + 1} dx = \frac{1}{2} \int \frac{du}{u} = \frac{1}{2} \log |u| + C_2 = \frac{1}{2} \log |x^2 + 1| + C_2. \]

### Step 3: Combine the Results
Now, we can combine the results from both integrals:

\[ \log |y| = \frac{1}{2} \log |x^2 + 1| + C, \]

where \(C = C_2 - C_1\) is a new constant.

### Step 4: Exponentiate to Solve for \(y\)
To solve for \(y\), we exponentiate both sides:

\[ |y| = e^{\frac{1}{2} \log |x^2 + 1| + C} = e^C \cdot (x^2 + 1)^{1/2}. \]

Let \(k = e^C\), which is a positive constant. Thus, we have:

\[ y = k \sqrt{x^2 + 1}. \]

### Step 5: Write the General Solution
The general solution of the differential equation is:

\[ y = C \sqrt{x^2 + 1}, \]

where \(C\) is a constant.

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