The differential equation of all non-horizontal lines in a plane is (a) `( b ) (c) (d)(( e ) (f) d^(( g )2( h ))( i ) y)/( j )(( k ) d (l) x^(( m )2( n ))( o ))( p ) (q) (r)` (s) (b) `( t ) (u) (v)(( w ) (x) d^(( y )2( z ))( a a ) x)/( b b )(( c c ) d (dd) y^(( e e )2( f f ))( g g ))( h h ) (ii)=0( j j )` (kk) (c) `( d ) (e) (f)(( g ) dy)/( h )(( i ) dx)( j ) (k)=0( l )` (m) (d) `( n ) (o) (p)(( q ) dx)/( r )(( s ) dy)( t ) (u)=0( v )` (w)

Options

A. `(d^(2)y)/(dx^(2))`
B. `(d^(2)x)/(dy^(2))=0`
C. `(dy)/(dx)=0`
D. `(dx)/(dy)=0`

Explanation

To solve the problem of finding the differential equation of all non-horizontal lines in a plane, we can follow these steps:

### Step 1: Understand the equation of a non-horizontal line
The general equation of a non-horizontal line in the xy-plane can be expressed as:
\[ Ax + By = C \]
where \( A \neq 0 \). This implies that the line has a slope and is not horizontal.

### Step 2: Rearranging the equation
We can rearrange the equation to express \( y \) in terms of \( x \):
\[ By = -Ax + C \]
\[ y = -\frac{A}{B}x + \frac{C}{B} \]

### Step 3: Differentiate the equation
To find the differential equation, we need to differentiate the equation with respect to \( x \):
\[ \frac{dy}{dx} = -\frac{A}{B} \]
This indicates that the slope of the line is constant and equal to \(-\frac{A}{B}\).

### Step 4: Differentiate again
Now, we differentiate again with respect to \( x \):
\[ \frac{d^2y}{dx^2} = 0 \]
This shows that the second derivative of \( y \) with respect to \( x \) is zero, which is a characteristic of linear functions (non-horizontal lines).

### Step 5: Write the final differential equation
Thus, the differential equation representing all non-horizontal lines in the plane is:
\[ \frac{d^2y}{dx^2} = 0 \]

### Conclusion
The correct answer is option (a):
\[ \frac{d^2y}{dx^2} = 0 \]

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