Question Number 26584 by gunawan last updated on 27/Dec/17
$$\int_{{a}} ^{{x}} \left({x}−{t}\right)^{\mathrm{5}} {y}\left({t}\right){dt}=\mathrm{4}{x}^{\mathrm{6}} \\ $$$${y}\left({x}\right)=… \\ $$
Commented by prakash jain last updated on 27/Dec/17
$$\frac{{d}}{{dx}}\int_{{a}\left({x}\right)} ^{{b}\left({x}\right)} {f}\left({x},{t}\right){dt} \\ $$$$={f}\left({x},{b}\left({x}\right)\right)\frac{{d}}{{dx}}{b}\left({x}\right)−{f}\left({x},{a}\left({x}\right)\right)\frac{{d}}{{dx}}{a}\left({x}\right)+ \\ $$$$\:\:\:\:\:\int_{{a}\left({x}\right)} ^{{b}\left({x}\right)} \frac{\partial}{\partial{x}}{f}\left({x},{t}\right){dt} \\ $$$$\frac{{d}}{{dx}}{a}\left({x}\right)=\mathrm{0} \\ $$$${f}\left({x},{x}\right)=\mathrm{0} \\ $$$$\mathrm{4}\centerdot\mathrm{6}\centerdot{x}^{\mathrm{5}} =\int_{{a}} ^{{x}} \mathrm{5}\left({x}−{t}\right)^{\mathrm{4}} {y}\left({t}\right){dt} \\ $$$$\mathrm{4}\centerdot\mathrm{6}\centerdot\mathrm{5}\centerdot{x}^{\mathrm{4}} =\int_{{a}} ^{{x}} \mathrm{5}\centerdot\mathrm{4}\left({x}−{t}\right)^{\mathrm{3}} {y}\left({t}\right){dt} \\ $$$${continue}\:{to}\:{get} \\ $$$$\mathrm{24}{x}=\int_{{a}} ^{{x}} {y}\left({t}\right){dt} \\ $$
Commented by prakash jain last updated on 27/Dec/17
$$\mathrm{Do}\:\mathrm{you}\:\mathrm{have}\:\mathrm{the}\:\mathrm{final}\:\mathrm{answer}? \\ $$$$\mathrm{i}\:\mathrm{am}\:\mathrm{not}\:\mathrm{able}\:\mathrm{to}\:\mathrm{get}\:\mathrm{to}\:\mathrm{a}\:\mathrm{final}\:\mathrm{answer}. \\ $$
Commented by prakash jain last updated on 27/Dec/17
$${F}\left({x}\right)=\int_{{a}} ^{{x}} \left({x}−{t}\right)^{\mathrm{5}} {y}\left({t}\right){dt}=\mathrm{4}{x}^{\mathrm{6}} \\ $$$${F}\left({a}\right)=\int_{{a}} ^{{a}} \left({a}−{t}\right)^{\mathrm{5}} {y}\left({t}\right){dt}=\mathrm{0} \\ $$$$\Rightarrow\mathrm{4}{a}^{\mathrm{6}} =\mathrm{0}\Rightarrow{a}=\mathrm{0} \\ $$$${can}\:{you}\:{please}\:{check}\:{question} \\ $$$${if}\:{a}=\mathrm{0},{y}=\mathrm{24} \\ $$
Commented by prakash jain last updated on 28/Dec/17
$$\mathrm{Integrate}\:\mathrm{after}\:\mathrm{puttimg}\:{y}=\mathrm{24}. \\ $$$$\mathrm{Also}\:\mathrm{value}\:\mathrm{of}\:{a}\:\mathrm{can}\:\mathrm{only}\:\mathrm{be}\:\mathrm{0}.\:\mathrm{Since} \\ $$$$\mathrm{F}\left({a}\right)=\mathrm{0}. \\ $$