Question Number 189066 by cortano12 last updated on 11/Mar/23 | ||
$$\:\mathrm{Given}\:\mathrm{f}\left(\mathrm{x}\right)+\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\left(\mathrm{x}+\mathrm{y}\right)^{\mathrm{2}} \:\mathrm{f}\left(\mathrm{y}\right)\:\mathrm{dy}=\mathrm{2x}^{\mathrm{2}} −\mathrm{3x}+\mathrm{1} \\ $$$$\:\mathrm{find}\:\mathrm{f}\left(\mathrm{x}\right). \\ $$ | ||
Answered by horsebrand11 last updated on 11/Mar/23 | ||
$$\mathrm{2}{x}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{1}={f}\left({x}\right)+\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\left({x}^{\mathrm{2}} +\mathrm{2}{xy}+{y}^{\mathrm{2}} \right){dy} \\ $$$$\mathrm{2}{x}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{1}={f}\left({x}\right)+{x}^{\mathrm{2}} \underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}{f}\left({y}\right){dy}+\mathrm{2}{x}\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}{y}\:{f}\left({y}\right){dy}+\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}{y}^{\mathrm{2}} \:{f}\left({y}\right)\:{dy} \\ $$$$\mathrm{let}\:\begin{cases}{{p}=\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}{f}\left({y}\right){dy}}\\{{q}=\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}{y}\:{f}\left({y}\right){dy}\:}\\{{r}=\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}{y}^{\mathrm{2}} \:{f}\left({y}\right){dy}\:}\end{cases} \\ $$$${f}\left({x}\right)=\mathrm{2}{x}^{\mathrm{2}} −{px}^{\mathrm{2}} −\mathrm{3}{x}−\mathrm{2}{qx}+\mathrm{1}−{r} \\ $$$${f}\left({x}\right)=\left(\mathrm{2}−{p}\right){x}^{\mathrm{2}} −\left(\mathrm{3}+\mathrm{2}{q}\right){x}+\left(\mathrm{1}−{r}\right) \\ $$$$ \\ $$ | ||
Answered by mr W last updated on 11/Mar/23 | ||
$${f}\left({x}\right)+\int_{\mathrm{0}} ^{\mathrm{1}} \left({x}+{y}\right)^{\mathrm{2}} {f}\left({y}\right){dy}=\mathrm{2}{x}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{1} \\ $$$${f}\left({x}\right)+\int_{\mathrm{0}} ^{\mathrm{1}} \left({x}^{\mathrm{2}} +\mathrm{2}{xy}+{y}^{\mathrm{2}} \right){f}\left({y}\right){dy}=\mathrm{2}{x}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{1} \\ $$$${f}\left({x}\right)+{x}^{\mathrm{2}} \int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({y}\right){dy}+\mathrm{2}{x}\int_{\mathrm{0}} ^{\mathrm{1}} {yf}\left({y}\right){dy}+\int_{\mathrm{0}} ^{\mathrm{1}} {y}^{\mathrm{2}} {f}\left({y}\right){dy}=\mathrm{2}{x}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{1} \\ $$$${f}\left({x}\right)+{Ax}^{\mathrm{2}} +\mathrm{2}{Bx}+{C}=\mathrm{2}{x}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{1} \\ $$$$\Rightarrow{f}\left({x}\right)=\left(\mathrm{2}−{A}\right){x}^{\mathrm{2}} −\left(\mathrm{3}+\mathrm{2}{B}\right){x}+\left(\mathrm{1}−{C}\right) \\ $$$${A}=\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({y}\right){dy}=\int_{\mathrm{0}} ^{\mathrm{1}} \left[\left(\mathrm{2}−{A}\right){y}^{\mathrm{2}} −\left(\mathrm{3}+\mathrm{2}{B}\right){y}+\left(\mathrm{1}−{C}\right)\right]{dy} \\ $$$${A}=\left(\mathrm{2}−{A}\right)\frac{\mathrm{1}}{\mathrm{3}}−\left(\mathrm{3}+\mathrm{2}{B}\right)\frac{\mathrm{1}}{\mathrm{2}}+\left(\mathrm{1}−{C}\right) \\ $$$$\Rightarrow\mathrm{8}{A}+\mathrm{6}{B}+\mathrm{6}{C}=\mathrm{1}\:\:\:...\left({i}\right) \\ $$$${B}=\int_{\mathrm{0}} ^{\mathrm{1}} {yf}\left({y}\right){dy}=\int_{\mathrm{0}} ^{\mathrm{1}} \left[\left(\mathrm{2}−{A}\right){y}^{\mathrm{3}} −\left(\mathrm{3}+\mathrm{2}{B}\right){y}^{\mathrm{2}} +\left(\mathrm{1}−{C}\right){y}\right]{dy} \\ $$$${B}=\left(\mathrm{2}−{A}\right)\frac{\mathrm{1}}{\mathrm{4}}−\left(\mathrm{3}+\mathrm{2}{B}\right)\frac{\mathrm{1}}{\mathrm{3}}+\left(\mathrm{1}−{C}\right)\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\Rightarrow\mathrm{3}{A}+\mathrm{20}{B}+\mathrm{6}{C}=\mathrm{0}\:\:\:...\left({ii}\right) \\ $$$${C}=\int_{\mathrm{0}} ^{\mathrm{1}} {y}^{\mathrm{2}} {f}\left({y}\right){dy}=\int_{\mathrm{0}} ^{\mathrm{1}} \left[\left(\mathrm{2}−{A}\right){y}^{\mathrm{4}} −\left(\mathrm{3}+\mathrm{2}{B}\right){y}^{\mathrm{3}} +\left(\mathrm{1}−{C}\right){y}^{\mathrm{2}} \right]{dy} \\ $$$${C}=\left(\mathrm{2}−{A}\right)\frac{\mathrm{1}}{\mathrm{5}}−\left(\mathrm{3}+\mathrm{2}{B}\right)\frac{\mathrm{1}}{\mathrm{4}}+\left(\mathrm{1}−{C}\right)\frac{\mathrm{1}}{\mathrm{3}} \\ $$$$\Rightarrow\mathrm{12}{A}+\mathrm{30}{B}+\mathrm{80}{C}=−\mathrm{1}\:\:\:...\left({iii}\right) \\ $$$${from}\:\left({i}\right),\left({ii}\right),\left({iii}\right): \\ $$$${A}=\frac{\mathrm{71}}{\mathrm{453}},{B}=−\frac{\mathrm{7}}{\mathrm{453}},{C}=−\frac{\mathrm{356}}{\mathrm{1359}} \\ $$$$\Rightarrow{f}\left({x}\right)=\frac{\mathrm{835}{x}^{\mathrm{2}} }{\mathrm{453}}−\frac{\mathrm{1345}{x}}{\mathrm{453}}+\frac{\mathrm{1715}}{\mathrm{1359}} \\ $$ | ||