Consider the family of all circles whose centers lie on the straight line `y=x` . If this family of circles is represented by the differential equation `P y^+Q y^(prime)+1=0,` where `P ,Q` are functions of `x , y` and `y^(prime)(h e r ey^(prime)=(dy)/(dx),y^=(d^2y)/(dx^2)),` then which of the following statements is (are) true? (a) `( b ) (c) P=y+x (d)` (e) (b) `( f ) (g) P=y-x (h)` (i) (c) `( d ) (e) P+Q=1-x+y+y +( f ) (g)(( h ) (i) y^(( j )prime( k ))( l ) ( m ))^(( n )2( o ))( p ) (q)` (r) (s) `( t ) (u) P-Q=x+y-y -( v ) (w)(( x ) (y) y^(( z )prime( a a ))( b b ) ( c c ))^(( d d )2( e e ))( f f ) (gg)` (hh)

Options

A. `P=y+x`
B. `P=y-x`
C. `P+Q=1-x+y+y^(')+(y^('))^(2)`
D. `P-Q=x+y-y^(')-(y^('))^(2)`

Explanation

To solve the problem, we need to derive the differential equation representing the family of circles whose centers lie on the line \( y = x \).

### Step-by-Step Solution:

1. **Equation of the Circle**:
The general equation of a circle with center \((h, k)\) and radius \(r\) is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Since the centers lie on the line \(y = x\), we can set \(h = k\). Thus, the equation becomes:
\[ (x - h)^2 + (y - h)^2 = r^2 \]

2. **Substituting \(h\)**:
Substitute \(h\) with \(y\) (since \(h = k = y\)):
\[ (x - y)^2 + (y - y)^2 = r^2 \]
Simplifying this gives:
\[ (x - y)^2 = r^2 \]

3. **Differentiating the Equation**:
Differentiate both sides with respect to \(x\):
\[ 2(x - y)(1 - \frac{dy}{dx}) = 0 \]
This implies:
\[ x - y = 0 \quad \text{or} \quad 1 - \frac{dy}{dx} = 0 \]
Hence, we have:
\[ \frac{dy}{dx} = 1 \quad \text{or} \quad x = y \]

4. **Finding \(h\)**:
Rearranging gives us:
\[ h = x - y \]

5. **Second Derivative**:
Now, differentiate again to find the second derivative:
\[ h \frac{d^2y}{dx^2} = 1 + y \frac{dy}{dx} + \left(\frac{dy}{dx}\right)^2 \]
Substituting \(h\):
\[ (x - y) \frac{d^2y}{dx^2} = 1 + y \frac{dy}{dx} + \left(\frac{dy}{dx}\right)^2 \]

6. **Rearranging**:
Rearranging gives us:
\[ (x - y) \frac{d^2y}{dx^2} - y \frac{dy}{dx} - \left(\frac{dy}{dx}\right)^2 - 1 = 0 \]

7. **Identifying \(P\) and \(Q\)**:
From the equation, we can identify:
\[ P = y - x \quad \text{and} \quad Q = 1 + \left(\frac{dy}{dx}\right) + \left(\frac{dy}{dx}\right)^2 \]

### Conclusion:
The correct statements based on the derived equations are:
- \(P = y - x\) (True)
- \(Q = 1 + \left(\frac{dy}{dx}\right) + \left(\frac{dy}{dx}\right)^2\) (True)