If `y=f(x)` satisfies the differential equation `(dy)/(dx)+(2x)/(1+x^(2))y=(3x^(2))/(1+x^(2))` where `f(1)=1`, then `f(2)` is equal to

Answer

To solve the differential equation

\[ \frac{dy}{dx} + \frac{2x}{1+x^2} y = \frac{3x^2}{1+x^2} \]

with the initial condition \( f(1) = 1 \), we will use the method of integrating factors.

### Step 1: Identify \( p(x) \) and \( q(x) \)

In the given equation, we can identify:
- \( p(x) = \frac{2x}{1+x^2} \)
- \( q(x) = \frac{3x^2}{1+x^2} \)

### Step 2: Find the integrating factor

The integrating factor \( \mu(x) \) is given by:

\[ \mu(x) = e^{\int p(x) \, dx} = e^{\int \frac{2x}{1+x^2} \, dx} \]

To solve the integral, we can use the substitution \( t = 1 + x^2 \), which gives \( dt = 2x \, dx \). Thus, the integral becomes:

\[ \int \frac{2x}{1+x^2} \, dx = \int \frac{1}{t} \, dt = \ln |t| + C = \ln(1+x^2) + C \]

So, the integrating factor is:

\[ \mu(x) = e^{\ln(1+x^2)} = 1+x^2 \]

### Step 3: Multiply through by the integrating factor

Now, we multiply the entire differential equation by the integrating factor:

\[ (1+x^2) \frac{dy}{dx} + (1+x^2) \frac{2x}{1+x^2} y = (1+x^2) \frac{3x^2}{1+x^2} \]

This simplifies to:

\[ (1+x^2) \frac{dy}{dx} + 2xy = 3x^2 \]

### Step 4: Rewrite the left-hand side

The left-hand side can be rewritten as the derivative of a product:

\[ \frac{d}{dx} \left( y(1+x^2) \right) = 3x^2 \]

### Step 5: Integrate both sides

Integrating both sides with respect to \( x \):

\[ y(1+x^2) = \int 3x^2 \, dx = x^3 + C \]

### Step 6: Solve for \( y \)

Thus, we have:

\[ y(1+x^2) = x^3 + C \]

So,

\[ y = \frac{x^3 + C}{1+x^2} \]

### Step 7: Use the initial condition to find \( C \)

We know that \( f(1) = 1 \):

\[ 1 = \frac{1^3 + C}{1 + 1^2} = \frac{1 + C}{2} \]

Multiplying both sides by 2 gives:

\[ 2 = 1 + C \implies C = 1 \]

### Step 8: Write the final solution

Now substituting \( C \) back into the equation for \( y \):

\[ y = \frac{x^3 + 1}{1+x^2} \]

### Step 9: Find \( f(2) \)

Now we need to find \( f(2) \):

\[ f(2) = \frac{2^3 + 1}{1 + 2^2} = \frac{8 + 1}{1 + 4} = \frac{9}{5} \]

### Final Answer

Thus, \( f(2) = \frac{9}{5} = 1.8 \).

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