Consider the family of all circles whose centers lie on the straight line `y=x`. If the family of circles is represented by the differential equation `Py^('')+Qy^(')+1=0`, where P, Q are functions of x, y and `y^(')` (here `y^(')=(dy)/(dx),y^('')=(d^(2)y)/(dx^(2))`, then which of the following statements is (are) true?

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)`

Answer

B, C

Explanation

The equation of the family of circles whose centres lie on the straight line y = x is
`x^(2)+y^(2)-2ax-2ay+c=0`, whose a and c are parameters.
Differentiating with respect to x, we obtain
`2x+2yy'-2a-2ay'=0rArra=(x+yy')/(1+y')`
Differentiating this with respect to x, we obtain
`((1+y')(1+(y')^(2)+yy')-(x+yy')y'')/((1+y')^(2))=0`
`rArr" "(1+y')(1+(y')^(2)+yy'')-(x+yy')y''=0`
`rArr" "1+y'((y')^(2)+y'+1)+y''(y-x)=0`
Comparing this equation with `Py''+Qy'+1=0`, we obtain
`P=y-x and Q=(y')^(2)+y'+1`
`rArr" "P=y-x and P+Q=1-x+y+y'+(y')^(2)`
Hence, options (b) and (c) are correct.