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Question Number 1603 by 123456 last updated on 25/Aug/15
((f(x))/(f′(x)))=((f′(x))/(f(x)))  f(x)=
$$\frac{{f}\left({x}\right)}{{f}'\left({x}\right)}=\frac{{f}'\left({x}\right)}{{f}\left({x}\right)} \\ $$$${f}\left({x}\right)= \\ $$
Answered by Rasheed Soomro last updated on 28/Aug/15
[ f ′(x) ]^2 =[ f(x) ]^2   f ′(x)=±f(x)  Let f(x)=y   (dy/dx)=±y⇒((1/y)) (dy/dx)=±1   ⇒∫[((1/y)) (dy/dx)]dx=∫(±1)dx  ln∣y∣+c=±x+c  ln∣y∣=±x+c  y=e^(±x+c) =e^(±x) .e^c =ke^(±x)   f(x)=ke^(±x)
$$\left[\:{f}\:'\left({x}\right)\:\right]^{\mathrm{2}} =\left[\:{f}\left({x}\right)\:\right]^{\mathrm{2}} \\ $$$${f}\:'\left({x}\right)=\pm{f}\left({x}\right) \\ $$$${Let}\:{f}\left({x}\right)={y}\: \\ $$$$\frac{{dy}}{{dx}}=\pm{y}\Rightarrow\left(\frac{\mathrm{1}}{{y}}\right)\:\frac{{dy}}{{dx}}=\pm\mathrm{1}\: \\ $$$$\Rightarrow\int\left[\left(\frac{\mathrm{1}}{{y}}\right)\:\frac{{dy}}{{dx}}\right]{dx}=\int\left(\pm\mathrm{1}\right){dx} \\ $$$${ln}\mid{y}\mid+{c}=\pm{x}+{c} \\ $$$${ln}\mid{y}\mid=\pm{x}+{c} \\ $$$${y}={e}^{\pm{x}+{c}} ={e}^{\pm{x}} .{e}^{{c}} ={ke}^{\pm{x}} \\ $$$${f}\left({x}\right)={ke}^{\pm{x}} \\ $$
Commented by 123456 last updated on 27/Aug/15
y=e^(±x+c) =e^(±x) e^c =ke^(±x)
$${y}={e}^{\pm{x}+{c}} ={e}^{\pm{x}} {e}^{{c}} ={ke}^{\pm{x}} \\ $$
Commented by Rasheed Soomro last updated on 27/Aug/15
Thanks for mentioning and correcting my mistake.  Pl continue same practice as I want to learn.
$$\boldsymbol{\mathrm{Thanks}}\:\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{mentioning}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{correcting}}\:\boldsymbol{\mathrm{my}}\:\boldsymbol{\mathrm{mistake}}. \\ $$$$\boldsymbol{\mathrm{Pl}}\:\boldsymbol{\mathrm{continue}}\:\boldsymbol{\mathrm{same}}\:\boldsymbol{\mathrm{practice}}\:\boldsymbol{\mathrm{as}}\:\boldsymbol{\mathrm{I}}\:\boldsymbol{\mathrm{want}}\:\boldsymbol{\mathrm{to}}\:\boldsymbol{\mathrm{learn}}. \\ $$

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