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To form the differential equation of the family of circles having a center on the y-axis and a radius of 3 units, we first need to express the equation of a circle with its center on the y-axis and radius 3 units.
The general equation of a circle with center (0, c) and radius r is given by:
x2+(y−c)2=r2x2+(y−c)2=r2
In this case, since the center lies on the y-axis, the x-coordinate of the center is 0, and the y-coordinate can be any value (let's denote it as cc).
Substituting cc with yy and rr with 33, we get:
x2+(y−y)2=32x2+(y−y)2=32 x2+y2=9x2+y2=9
Now, to form the differential equation, we'll differentiate both sides with respect to xx:
ddx(x2)+ddx(y2)=ddx(9)dxd(x2)+dxd(y2)=dxd(9) 2x+2ydydx=02x+2ydxdy=0
This can be simplified to:
x+ydydx=0x+ydxdy=0
And finally, solving for dydxdxdy, we get the differential equation:
dydx=−xydxdy=−yx
This is the differential equation of the family of circles with centers on the y-axis and radius 3 units.
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