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Find lambda: y(x)=\lambda \int _{0}^{1}xty(t)dt\, if y(x)=x\,

If y(x)=x\,, then

y(x)=\lambda \int _{0}^{1}xty(t)\ dt\, \Longrightarrow x=\lambda \int _{0}^{1}xtt\ dt\,
\Longrightarrow x=x\lambda \int _{0}^{1}t^{2}\ dt\,
\Longrightarrow 1=\lambda \left({\frac  {1}{3}}t^{3}\left.\right|_{0}^{1}\right)\,
\Longrightarrow 1={\frac  {\lambda }{3}}\,

Therefore, \lambda =3\,.


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