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Show that, v = A/r + B satisfies the di...

Show that, `v = A/r + B` satisfies the differential equation `(d^2v)/(dr^2)+2/r.(dv)/(dr)=0`

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The correct Answer is:
Since the given relation contains two arbitrary constants, we differentiate it two times w.r.t.r, and eliminate A and B.
`v=A/r+Bimplies(dv)/(dr)=(-A)/(r^(2))" "...(i)`
`implies(d^(2)v)/(dr^(2))=(2A)/(r^(3))" "...(ii)`
On dividing (ii) by (i), we get
`((d^(2)v//dr^(2)))/((dv//dr))={(2A)/(r^(3))xx(r^(2))/((-A))}=(-2)/(r)`
`implies(d^(2)v)/(dr^(2))=(-2)/(r).(dv)/(dr)`
`implies(d^(2)v)/(dr^(2))+2/r.(dv)/(dr)=0.`
Hence, `v=A/r+B` is a solution of the differential equation `(d^(2)v)/(dr^(2))+2/r.(dv)/(dr)=0.`
Since the given relation contains two arbitrary constants, we differentiate it two times w.r.t.r, and eliminate A and B.
`v=A/r+Bimplies(dv)/(dr)=(-A)/(r^(2))" "...(i)`
`implies(d^(2)v)/(dr^(2))=(2A)/(r^(3))" "...(ii)`
On dividing (ii) by (i), we get
`((d^(2)v//dr^(2)))/((dv//dr))={(2A)/(r^(3))xx(r^(2))/((-A))}=(-2)/(r)`
`implies(d^(2)v)/(dr^(2))=(-2)/(r).(dv)/(dr)`
`implies(d^(2)v)/(dr^(2))+2/r.(dv)/(dr)=0.`
Hence, `v=A/r+B` is a solution of the differential equation `(d^(2)v)/(dr^(2))+2/r.(dv)/(dr)=0.`
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