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Find the differential equation representing the family of curves v=A/r+ B, where A and B are arbitrary constants. - Mathematics

Find the differential equation representing the family of curves v=A/r+ B, where A and B are arbitrary constants.

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Solution

 

The equation of the family of curves is v=A/r+B, where A and B are arbitrary constants.

We have

v=Ar+B

Differentiating both sides with respect to r, we get

`(dv)/(dr)=-A/r^2+0`

`⇒r^2(dv)/(dr)=−A`

Again, differentiating both sides with respect to r, we get

`r^2xx(d^2v)/(d^2r)+2rxx(dv)/(dr)=0`

`⇒r(d^2v)/(d^2r)+2(dv)/(dr)=0`

This is the differential equation representing the family of the given curves

 
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