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1)

 Consider the family of all circles whose centres lie on the straight line y=x, If this family of circles is represented by the differential equation   Py"+Qy+1=0 , where P,Q are the functions of x, y and y'   (here,    y=dydx,y=d2ydx2), then which of the following statement (s) is /are true?


A) P=y+x

B) P=y-x

C) P+Q=1x+y+y+(y)2

D) PQ=x+yy(y)2

Answer:

Option B,C

Explanation:

 Since the centre lies on y=x.

        Equation of circle is

                    x^{2}+y^{2}-2ax-2ay+c=0

  On differentiating , we get

          2x+2yy'-2a-2ay'=0

   \Rightarrow       x+yy'-a-ay'=0

  \Rightarrow    a=\frac{x+yy'}{1+y'}

 Again differentiating , we get

         0  = \frac{(1+y')[1+yy'+(y')^{2}]-(x+yy').(y")}{(1+y')^{2}}

\Rightarrow     (1+y')[1+(y')^{2}+yy"]- (x+yy')(y")=0

\Rightarrow    1+y'[(y')^{2}+y'+1]+y"(y-x)=0

 On comparing  with   Py"+Qy'+1=0, we get 

 P=y-x  and Q= (y')^{2}+y'+1