Question Number 1088 by 112358 last updated on 10/Jun/15
$${Solve}\:{the}\:{following}\:{integral} \\ $$$${equation}\:{for}\:{f}\left({x}\right): \\ $$$$\int_{\mathrm{0}} ^{\:{x}} {f}\left({t}\right){dt}=\mathrm{3}{f}\left({x}\right)+{k} \\ $$$${where}\:{k}\:{is}\:{a}\:{constant}.\: \\ $$
Answered by prakash jain last updated on 10/Jun/15
$${f}\left({x}\right)=\mathrm{3}{f}\:'\left({x}\right) \\ $$$${f}\left({x}\right)={c}\:{e}^{{x}/\mathrm{3}} \\ $$$$\int_{\mathrm{0}} ^{\:{x}} {f}\left({t}\right){dt}={c}\int_{\mathrm{0}} ^{\:{x}} {e}^{{t}/\mathrm{3}} {dt}=\mathrm{3}{ce}^{{x}/\mathrm{3}} −\mathrm{3}{c}=\mathrm{3}{ce}^{{x}/\mathrm{3}} +{k} \\ $$$$\Rightarrow{c}=−\frac{{k}}{\mathrm{3}} \\ $$$${f}\left({x}\right)=−\frac{{k}}{\mathrm{3}}{e}^{{x}/\mathrm{3}} \\ $$
Commented by 112358 last updated on 11/Jun/15
$${thanks} \\ $$