Solve the differential equation `(x+1)(dy)/(dx) =2xy, " given that " y(2)=3.`

Explanation

To solve the differential equation \((x+1)\frac{dy}{dx} = 2xy\) given that \(y(2) = 3\), we will follow these steps:

### Step 1: Rearranging the Equation
We start by rearranging the given differential equation to separate the variables \(y\) and \(x\).

\[ \frac{dy}{dx} = \frac{2xy}{x+1} \]

### Step 2: Separating Variables
Next, we separate the variables by dividing both sides by \(y\) and multiplying both sides by \(dx\):

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

### Step 3: Integrating Both Sides
Now, we integrate both sides. The left side integrates to \(\ln |y|\), and for the right side, we will need to simplify the expression before integrating.

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

To integrate the right side, we can rewrite \(\frac{2x}{x+1}\) as:

\[ \frac{2x}{x+1} = 2 - \frac{2}{x+1} \]

Thus, we can integrate:

\[ \int \frac{1}{y} dy = \int \left(2 - \frac{2}{x+1}\right) dx \]

This gives:

\[ \ln |y| = 2x - 2\ln |x+1| + C \]

### Step 4: Simplifying the Equation
We can exponentiate both sides to eliminate the logarithm:

\[ y = e^{2x - 2\ln |x+1| + C} \]

This can be rewritten as:

\[ y = e^C \cdot e^{2x} \cdot (x+1)^{-2} \]

Let \(k = e^C\), then:

\[ y = k \cdot \frac{e^{2x}}{(x+1)^2} \]

### Step 5: Applying Initial Condition
Now we apply the initial condition \(y(2) = 3\):

\[ 3 = k \cdot \frac{e^{4}}{(2+1)^2} \]

This simplifies to:

\[ 3 = k \cdot \frac{e^{4}}{9} \]

Thus,

\[ k = \frac{27}{e^{4}} \]

### Step 6: Final Solution
Substituting \(k\) back into the equation for \(y\):

\[ y = \frac{27}{e^{4}} \cdot \frac{e^{2x}}{(x+1)^2} \]

This can be simplified to:

\[ y = \frac{27 e^{2x}}{(x+1)^2 e^{4}} = \frac{27 e^{2x - 4}}{(x+1)^2} \]

### Final Answer
Thus, the solution to the differential equation is:

\[ y = \frac{27 e^{2x - 4}}{(x+1)^2} \]