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Question Number 13751 by Nayon last updated on 23/May/17

(ds/dt)=v,(dv/dt)=a,(da/dt)=b,(db/dt)=e,(de/dt)=f  (df/dt)=g,(dg/dt)=h,(dh/dt)=i,(di/dt)=j,(dj/dt)=k,.....  now if we continue this process to  infinity..and if v_0 ,v,a,b,e,f,g,h,i,  j,................=1 .then calculate  the formula of v and s ...

$$\frac{{ds}}{{dt}}={v},\frac{{dv}}{{dt}}={a},\frac{{da}}{{dt}}={b},\frac{{db}}{{dt}}={e},\frac{{de}}{{dt}}={f} \\ $$$$\frac{{df}}{{dt}}={g},\frac{{dg}}{{dt}}={h},\frac{{dh}}{{dt}}={i},\frac{{di}}{{dt}}={j},\frac{{dj}}{{dt}}={k},..... \\ $$$${now}\:{if}\:{we}\:{continue}\:{this}\:{process}\:{to} \\ $$$${infinity}..{and}\:{if}\:{v}_{\mathrm{0}} ,{v},{a},{b},{e},{f},{g},{h},{i}, \\ $$$${j},................=\mathrm{1}\:.{then}\:{calculate} \\ $$$${the}\:{formula}\:{of}\:{v}\:{and}\:{s}\:... \\ $$$$ \\ $$

Commented by ajfour last updated on 23/May/17

s=v=e^t

$${s}={v}=\boldsymbol{{e}}^{\boldsymbol{{t}}} \\ $$

Commented by Nayon last updated on 23/May/17

detAils

$${detAils} \\ $$

Commented by Nayon last updated on 23/May/17

v=e^t  right but  s=e^t −1

$${v}={e}^{{t}} \:{right}\:{but} \\ $$$${s}={e}^{{t}} −\mathrm{1} \\ $$$$ \\ $$

Commented by ajfour last updated on 23/May/17

agree

$${agree} \\ $$

Answered by ajfour last updated on 23/May/17

s(t)=s(0)+ts′(0)+(t^2 /(2!))s′′(0)+...  and in question,  s′(0)=s′′(0)=....=1  ,  s(0)=0  so,  s(t)=−1+(1+t+(t^2 /(2!))+....)          s(t)=e^t −1          v(t)=(ds/dt)=e^t   .

$${s}\left({t}\right)={s}\left(\mathrm{0}\right)+{ts}'\left(\mathrm{0}\right)+\frac{{t}^{\mathrm{2}} }{\mathrm{2}!}{s}''\left(\mathrm{0}\right)+... \\ $$$${and}\:{in}\:{question}, \\ $$$${s}'\left(\mathrm{0}\right)={s}''\left(\mathrm{0}\right)=....=\mathrm{1}\:\:,\:\:{s}\left(\mathrm{0}\right)=\mathrm{0} \\ $$$${so},\:\:{s}\left({t}\right)=−\mathrm{1}+\left(\mathrm{1}+{t}+\frac{{t}^{\mathrm{2}} }{\mathrm{2}!}+....\right) \\ $$$$\:\:\:\:\:\:\:\:{s}\left({t}\right)={e}^{{t}} −\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\:{v}\left({t}\right)=\frac{{ds}}{{dt}}={e}^{{t}} \:\:. \\ $$$$ \\ $$

Commented by Nayon last updated on 24/May/17

Has there any object in the   universe which adhere this formula?

$${Has}\:{there}\:{any}\:{object}\:{in}\:{the}\: \\ $$$${universe}\:{which}\:{adhere}\:{this}\:{formula}? \\ $$

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