VC2.14

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Multiply both sides of the given equation by vector 2\frac{dr}{dt}\, we have

2\frac{dr}{dt}\cdot \frac{d^2 r}{dt^2}=-2n^2 r\cdot \frac{dr}{dt}\,

Let this be Equation 1.

But \frac{d}{dt}[\frac{dr}{dt}]^2=\frac{d}{dt}[\frac{dr}{dt}\cdot\frac{dr}{dt}]=\frac{d^2 r}{dt^2}\cdot\frac{dr}{dt^2}+\frac{dr}{dt}\cdot\frac{d^2 r}{dt^2}\,

Hence 2\frac{dr}{dt}\cdot\frac{d^2 r}{dt^2}=\frac{d}{dt}[\frac{dr}{dt}]^2\, [Equation 2]

Again \frac{d}{dt}(r^2)=\frac{d}{dt}(r\cdot r)=\frac{dr}{dt}\cdot r+r\cdot \frac{dr}{dt}\,

Hence 2r\cdot \frac{dr}{dt}=\frac{d}{dt}(r^2)\, [Equation 3]

Using 2 & 3,(1) reduces to \frac{d}{dt}[\frac{dr}{dt}]^2=-n^2\frac{d}{dt}(r^2)\,

Integrating this,we get [\frac{dr}{dt}]^2=-n^2 r^2+c\,

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