Menu Close

Solve-the-following-integral-equation-for-f-x-0-x-f-t-dt-3f-x-k-where-k-is-a-constant-




Question Number 1088 by 112358 last updated on 10/Jun/15
Solve the following integral  equation for f(x):  ∫_0 ^( x) f(t)dt=3f(x)+k  where k is a constant.
$${Solve}\:{the}\:{following}\:{integral} \\ $$$${equation}\:{for}\:{f}\left({x}\right): \\ $$$$\int_{\mathrm{0}} ^{\:{x}} {f}\left({t}\right){dt}=\mathrm{3}{f}\left({x}\right)+{k} \\ $$$${where}\:{k}\:{is}\:{a}\:{constant}.\: \\ $$
Answered by prakash jain last updated on 10/Jun/15
f(x)=3f ′(x)  f(x)=c e^(x/3)   ∫_0 ^( x) f(t)dt=c∫_0 ^( x) e^(t/3) dt=3ce^(x/3) −3c=3ce^(x/3) +k  ⇒c=−(k/3)  f(x)=−(k/3)e^(x/3)
$${f}\left({x}\right)=\mathrm{3}{f}\:'\left({x}\right) \\ $$$${f}\left({x}\right)={c}\:{e}^{{x}/\mathrm{3}} \\ $$$$\int_{\mathrm{0}} ^{\:{x}} {f}\left({t}\right){dt}={c}\int_{\mathrm{0}} ^{\:{x}} {e}^{{t}/\mathrm{3}} {dt}=\mathrm{3}{ce}^{{x}/\mathrm{3}} −\mathrm{3}{c}=\mathrm{3}{ce}^{{x}/\mathrm{3}} +{k} \\ $$$$\Rightarrow{c}=−\frac{{k}}{\mathrm{3}} \\ $$$${f}\left({x}\right)=−\frac{{k}}{\mathrm{3}}{e}^{{x}/\mathrm{3}} \\ $$
Commented by 112358 last updated on 11/Jun/15
thanks
$${thanks} \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *