The solution of `(1+x)(dy)/(dx)+1=e^(x-y)` is

Options

A. `e^(y)(x+1)=c`
B. `e^(y)(x+1)=e^(x)+c`
C. `e^(y)(x+1)=ce^(x)`
D. `(x+1)=e^(x)+c`

Explanation

To solve the differential equation \((1+x)\frac{dy}{dx} + 1 = e^{x-y}\), we will follow these steps:

### Step 1: Rearranging the Equation
First, we can rearrange the equation to isolate \(\frac{dy}{dx}\):
\[ (1+x)\frac{dy}{dx} = e^{x-y} - 1 \]
Now, we can express \(\frac{dy}{dx}\) as:
\[ \frac{dy}{dx} = \frac{e^{x-y} - 1}{1+x} \]

**Hint:** Rearranging the equation helps to isolate the derivative on one side.

### Step 2: Substituting for \(e^{x-y}\)
Next, we can rewrite \(e^{x-y}\) as:
\[ e^{x-y} = \frac{e^x}{e^y} \]
Substituting this into our equation gives:
\[ \frac{dy}{dx} = \frac{\frac{e^x}{e^y} - 1}{1+x} \]

**Hint:** Using properties of exponents can simplify the expression.

### Step 3: Multiplying Both Sides by \(e^y\)
To eliminate the fraction, we can multiply both sides by \(e^y(1+x)\):
\[ e^y(1+x)\frac{dy}{dx} = e^x - e^y(1+x) \]

**Hint:** Multiplying by a common factor can help simplify the equation.

### Step 4: Rearranging and Integrating
Now, we can rearrange the equation:
\[ e^y(1+x)\frac{dy}{dx} + e^y(1+x) = e^x \]
This can be expressed as:
\[ e^y dy + (1+x)e^y dy = e^x dx \]
Now we can integrate both sides:
\[ \int e^y dy + \int (1+x)e^y dy = \int e^x dx \]

**Hint:** Setting up for integration requires careful rearrangement.

### Step 5: Performing the Integrals
The integrals yield:
\[ e^y + (1+x)e^y = e^x + C \]
This simplifies to:
\[ e^y(1+x) = e^x + C \]

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

### Step 6: Final Solution
To express \(y\) explicitly, we can take the natural logarithm:
\[ y = \ln\left(\frac{e^x + C}{1+x}\right) \]

**Hint:** Using logarithmic properties can help isolate \(y\).

### Summary
The solution to the differential equation \((1+x)\frac{dy}{dx} + 1 = e^{x-y}\) is:
\[ y = \ln\left(\frac{e^x + C}{1+x}\right) \]