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Question

Show that the differential equation of the family of circles having their centre at the origin and radius a is
x+ydydx=0

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Solution

y=eacos1x
1x1
y1=dydx=ddx(eacos1x)=eacos1x.1.a1x2=a1x2eacos1x
y2=d2ydx2=dy1dx=ddx{a1x2eacos1x}
=a1x2.eacos1x.a1x2+eacos1x.(a){a2x21x2}1x2
=a2(1x2)eacos1xax(1x2)3/2eacos1x
(1x2)y2=a2eacos1xax1x2eacos1x
xy1=ax1x2eacos1x
a2y=a2eacos1x
(1x2)y2xy1a2y=a2eacos1xax1x2eacos1x+ax1x2eacos1x
a2eacos1x
=0
(1x2)y2xy1a2y=0 (proved)

1107669_1139500_ans_55b6c6b285d54faebc4e0cf4b74e176d.jpg

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