The differential equation `(dy)/(dx)=(x+y-1)/(x+y+1)`
reduces to variable separable form by making the substitution

Options

A. `x+y=v`
B. `x-y=v`
C. `y=vx`
D. `x=vy`

Explanation

To solve the differential equation \(\frac{dy}{dx} = \frac{x+y-1}{x+y+1}\) and reduce it to a variable separable form, we will use the substitution \(x + y = v\). Here are the steps to arrive at the solution:

### Step 1: Make the substitution
Let:
\[ v = x + y \]
This implies:
\[ y = v - x \]

### Step 2: Differentiate both sides
Now, differentiate both sides with respect to \(x\):
\[ \frac{dy}{dx} = \frac{dv}{dx} - 1 \]

### Step 3: Substitute into the original equation
Substituting \(y\) and \(\frac{dy}{dx}\) into the original differential equation:
\[ \frac{dv}{dx} - 1 = \frac{x + (v - x) - 1}{x + (v - x) + 1} \]
This simplifies to:
\[ \frac{dv}{dx} - 1 = \frac{v - 1}{v + 1} \]

### Step 4: Rearrange the equation
Rearranging gives:
\[ \frac{dv}{dx} = \frac{v - 1}{v + 1} + 1 \]
Now, simplifying the right-hand side:
\[ \frac{dv}{dx} = \frac{v - 1 + v + 1}{v + 1} = \frac{2v}{v + 1} \]

### Step 5: Separate the variables
Now, we can separate the variables:
\[ \frac{v + 1}{2v} dv = dx \]

### Step 6: Integrate both sides
Now, we can integrate both sides:
\[ \int \frac{v + 1}{2v} dv = \int dx \]

### Step 7: Solve the integrals
The left side can be split into two integrals:
\[ \int \left(\frac{1}{2} + \frac{1}{2v}\right) dv = \int dx \]
This gives:
\[ \frac{1}{2}v + \frac{1}{2} \ln |v| = x + C \]

### Conclusion
Thus, the substitution \(x + y = v\) reduces the given differential equation to a variable separable form.

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