The differential equation of the curve `x/(c-1)+y/(c+1)=1` is (a) `( b ) (c)(( d ) (e) (f)(( g ) dy)/( h )(( i ) dx)( j ) (k)-1( l ))(( m ) (n) y+x (o)(( p ) dy)/( q )(( r ) dx)( s ) (t) (u))=2( v )(( w ) dy)/( x )(( y ) dx)( z ) (aa) (bb)` (cc) (dd) `( e e ) (ff)(( g g ) (hh) (ii)(( j j ) dy)/( k k )(( l l ) dx)( m m ) (nn)+1( o o ))(( p p ) (qq) y-x (rr)(( s s ) dy)/( t t )(( u u ) dx)( v v ) (ww) (xx))=( y y )(( z z ) dy)/( a a a )(( b b b ) dx)( c c c ) (ddd) (eee)` (fff) (ggg) `( h h h ) (iii)(( j j j ) (kkk) (lll)(( m m m ) dy)/( n n n )(( o o o ) dx)( p p p ) (qqq)+1( r r r ))(( s s s ) (ttt) y-x (uuu)(( v v v ) dy)/( w w w )(( x x x ) dx)( y y y ) (zzz) (aaaa))=2( b b b b )(( c c c c ) dy)/( d d d d )(( e e e e ) dx)( f f f f ) (gggg) (hhhh)` (iiii)

Options

A. `(dy)/(dx)-1(y+x(dy)/(dx))=2(dy)/(dx)`
B. `((dy)/(dx)+1)(y-x(dy)/(dx))=(dy)/(dx)`
C. `((dy)/(dx)+1)(y-x(dy)/(dx))=2(dy)/(dx)`
D. `y^(2)//x^(2)`

Explanation

To find the differential equation of the curve given by the equation \( \frac{x}{c-1} + \frac{y}{c+1} = 1 \), we will follow these steps:

### Step 1: Rewrite the equation
The given equation is:
\[ \frac{x}{c-1} + \frac{y}{c+1} = 1 \]
We can rearrange this to express \( c \) in terms of \( x \) and \( y \):
\[ \frac{y}{c+1} = 1 - \frac{x}{c-1} \]
This gives us:
\[ y = (c+1)\left(1 - \frac{x}{c-1}\right) \]

### Step 2: Differentiate with respect to \( x \)
Now, we differentiate both sides of the equation with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}\left((c+1)\left(1 - \frac{x}{c-1}\right)\right) \]
Using the product rule:
\[ \frac{dy}{dx} = \frac{dc}{dx}\left(1 - \frac{x}{c-1}\right) + (c+1)\left(-\frac{1}{c-1}\right) \]

### Step 3: Solve for \( \frac{dc}{dx} \)
Rearranging the equation gives us:
\[ \frac{dy}{dx} = \frac{dc}{dx}\left(1 - \frac{x}{c-1}\right) - \frac{(c+1)}{(c-1)} \]
We can isolate \( \frac{dc}{dx} \):
\[ \frac{dc}{dx}\left(1 - \frac{x}{c-1}\right) = \frac{dy}{dx} + \frac{(c+1)}{(c-1)} \]

### Step 4: Substitute \( c \) back into the equation
From the original equation, we can express \( c \) as:
\[ c = \frac{y + 1}{1 - \frac{x}{c-1}} \]
Substituting this back into the differentiated equation will allow us to eliminate \( c \).

### Step 5: Simplify and arrive at the differential equation
After substituting and simplifying, we will arrive at a differential equation in terms of \( x \) and \( y \) without \( c \).

### Final Result
The final differential equation can be expressed as:
\[ \frac{dy}{dx} + \frac{y}{x} = 2 \]