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Question

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 Py′′+Qy+1=0, where P,Q are functions of x,y and y( here y=dydx,y′′=d2ydx2), then which of the following statements is (are) true ?

A
P=y+x
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B
P=yx
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C
P+Q=1x+y+y+(y)2
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D
PQ=x+yy(y)2
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Solution

The correct options are
B P=yx
C P+Q=1x+y+y+(y)2
As the center of the circle lies on the line y=x,
So equation of the circle is
(xh)2+(yh)2=r2
Now differentiate w.r.t. x, we get
2x2h+2(yh)y=02x+2yy2h2hy=0h=x+yy1+y (1)
Again differentiate w.r.t. x, we get
2+2(y)2+2yy′′2hy′′=0
Now putting the value of h, we get
2+2(y)2+2yy′′2(x+yy1+y)y′′=01+y+(y)2+(y)3+yy′′+yyy′′xy′′yyy′′=0(yx)y′′+(1+y+(y)2)y+1=0(Py′′+Qy+1=0) where P=yx , Q=1+y+(y)2

P+Q=1x+y+y+(y)2 and P=yx are the correct options.

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