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Question Number 26584 by gunawan last updated on 27/Dec/17

∫_a ^x (x−t)^5 y(t)dt=4x^6 y(x)=...

$$\int_{{a}} ^{{x}} \left({x}−{t}\right)^{\mathrm{5}} {y}\left({t}\right){dt}=\mathrm{4}{x}^{\mathrm{6}} \\ $$$${y}\left({x}\right)=... \\ $$

Commented by prakash jain last updated on 27/Dec/17

(d/dx)∫_(a(x)) ^(b(x)) f(x,t)dt =f(x,b(x))(d/dx)b(x)−f(x,a(x))(d/dx)a(x)+ ∫_(a(x)) ^(b(x)) (∂/∂x)f(x,t)dt (d/dx)a(x)=0 f(x,x)=0 4∙6∙x^5 =∫_a ^x 5(x−t)^4 y(t)dt 4∙6∙5∙x^4 =∫_a ^x 5∙4(x−t)^3 y(t)dt continue to get 24x=∫_a ^x y(t)dt

$$\frac{{d}}{{dx}}\int_{{a}\left({x}\right)} ^{{b}\left({x}\right)} {f}\left({x},{t}\right){dt} \\ $$$$={f}\left({x},{b}\left({x}\right)\right)\frac{{d}}{{dx}}{b}\left({x}\right)−{f}\left({x},{a}\left({x}\right)\right)\frac{{d}}{{dx}}{a}\left({x}\right)+ \\ $$$$\:\:\:\:\:\int_{{a}\left({x}\right)} ^{{b}\left({x}\right)} \frac{\partial}{\partial{x}}{f}\left({x},{t}\right){dt} \\ $$$$\frac{{d}}{{dx}}{a}\left({x}\right)=\mathrm{0} \\ $$$${f}\left({x},{x}\right)=\mathrm{0} \\ $$$$\mathrm{4}\centerdot\mathrm{6}\centerdot{x}^{\mathrm{5}} =\int_{{a}} ^{{x}} \mathrm{5}\left({x}−{t}\right)^{\mathrm{4}} {y}\left({t}\right){dt} \\ $$$$\mathrm{4}\centerdot\mathrm{6}\centerdot\mathrm{5}\centerdot{x}^{\mathrm{4}} =\int_{{a}} ^{{x}} \mathrm{5}\centerdot\mathrm{4}\left({x}−{t}\right)^{\mathrm{3}} {y}\left({t}\right){dt} \\ $$$${continue}\:{to}\:{get} \\ $$$$\mathrm{24}{x}=\int_{{a}} ^{{x}} {y}\left({t}\right){dt} \\ $$

Commented by prakash jain last updated on 27/Dec/17

Do you have the final answer? i am not able to get to a final answer.

$$\mathrm{Do}\:\mathrm{you}\:\mathrm{have}\:\mathrm{the}\:\mathrm{final}\:\mathrm{answer}? \\ $$$$\mathrm{i}\:\mathrm{am}\:\mathrm{not}\:\mathrm{able}\:\mathrm{to}\:\mathrm{get}\:\mathrm{to}\:\mathrm{a}\:\mathrm{final}\:\mathrm{answer}. \\ $$

Commented by prakash jain last updated on 27/Dec/17

F(x)=∫_a ^x (x−t)^5 y(t)dt=4x^6 F(a)=∫_a ^a (a−t)^5 y(t)dt=0 ⇒4a^6 =0⇒a=0 can you please check question if a=0,y=24

$${F}\left({x}\right)=\int_{{a}} ^{{x}} \left({x}−{t}\right)^{\mathrm{5}} {y}\left({t}\right){dt}=\mathrm{4}{x}^{\mathrm{6}} \\ $$$${F}\left({a}\right)=\int_{{a}} ^{{a}} \left({a}−{t}\right)^{\mathrm{5}} {y}\left({t}\right){dt}=\mathrm{0} \\ $$$$\Rightarrow\mathrm{4}{a}^{\mathrm{6}} =\mathrm{0}\Rightarrow{a}=\mathrm{0} \\ $$$${can}\:{you}\:{please}\:{check}\:{question} \\ $$$${if}\:{a}=\mathrm{0},{y}=\mathrm{24} \\ $$

Commented by prakash jain last updated on 28/Dec/17

Integrate after puttimg y=24. Also value of a can only be 0. Since F(a)=0.

$$\mathrm{Integrate}\:\mathrm{after}\:\mathrm{puttimg}\:{y}=\mathrm{24}. \\ $$$$\mathrm{Also}\:\mathrm{value}\:\mathrm{of}\:{a}\:\mathrm{can}\:\mathrm{only}\:\mathrm{be}\:\mathrm{0}.\:\mathrm{Since} \\ $$$$\mathrm{F}\left({a}\right)=\mathrm{0}. \\ $$

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