If `y(x)` is a solution of `(dy)/(dx)-(xy)/(1+x)=(1)/(1+x)` and `y(0)=-1`, then the value of `y(2)` is

Options

A. `-(1)/(2)`
B. `-(1)/(3)`
C. `-(1)/(4)`
D. `-(1)/(5)`

Answer

To solve the differential equation given by

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

with the initial condition \(y(0) = -1\), we will follow these steps:

### Step 1: Identify the form of the differential equation
This is a first-order linear differential equation of the form

\[ \frac{dy}{dx} + P(x)y = Q(x) \]

where \(P(x) = -\frac{x}{1+x}\) and \(Q(x) = \frac{1}{1+x}\).

**Hint:** Recognize that the equation can be rearranged to fit the standard form of a linear differential equation.

### Step 2: Find the integrating factor
The integrating factor \(IF\) is given by

\[ IF = e^{\int P(x) \, dx} = e^{\int -\frac{x}{1+x} \, dx} \]

To solve the integral, we can simplify it:

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

This gives us:

\[ -x + \ln(1+x) \]

Thus, the integrating factor becomes:

\[ IF = e^{-x + \ln(1+x)} = (1+x)e^{-x} \]

**Hint:** When finding the integrating factor, remember to simplify the integral carefully.

### Step 3: Multiply the entire differential equation by the integrating factor
We multiply the original equation by the integrating factor:

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

This simplifies to:

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

**Hint:** Multiplying by the integrating factor allows us to express the left-hand side as a derivative.

### Step 4: Rewrite the left-hand side as a derivative
The left-hand side can be rewritten as:

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

**Hint:** Recognize that the left-hand side is the derivative of a product.

### Step 5: Integrate both sides
Integrating both sides gives:

\[ y(1+x)e^{-x} = \int e^{-x} \, dx = -e^{-x} + C \]

**Hint:** Remember to include the constant of integration after integrating.

### Step 6: Solve for \(y\)
Now, we can solve for \(y\):

\[ y(1+x) = -1 + Ce^{x} \]

Thus,

\[ y = \frac{-1 + Ce^{x}}{1+x} \]

**Hint:** Isolate \(y\) to express it in terms of \(C\).

### Step 7: Apply the initial condition
Using the initial condition \(y(0) = -1\):

\[ -1 = \frac{-1 + C}{1+0} \]

This simplifies to:

\[ -1 = -1 + C \implies C = 0 \]

### Step 8: Write the particular solution
Thus, the particular solution is:

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

**Hint:** Substitute the initial condition carefully to find the constant.

### Step 9: Find \(y(2)\)
Now, we need to find \(y(2)\):

\[ y(2) = \frac{-1}{1+2} = \frac{-1}{3} \]

### Final Answer
Thus, the value of \(y(2)\) is

\[ \boxed{-\frac{1}{3}} \]