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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

                    x2+y22ax2ay+c=0

  On differentiating , we get

          2x+2yy2a2ay=0

          x+yyaay=0

      a=x+yy1+y

 Again differentiating , we get

         0  =(1+y)[1+yy+(y)2](x+yy).(y")(1+y)2

    (1+y)[1+(y)2+yy"](x+yy)(y")=0

    1+y[(y)2+y+1]+y"(yx)=0

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

 P=y-x  and Q=(y)2+y+1