IE18

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Solve: \sin(x)=\int _{0}^{x}e^{{x-t}}u(t)dt\,

Differentiate both sides with respect to x.

\cos(x)={\frac  {d}{dx}}\int _{0}^{x}e^{{x-t}}u(t)dt\,

\cos(x)=\int _{0}^{x}{\frac  {\partial }{\partial x}}e^{{x-t}}u(t)dt+u(x)\,

\cos(x)=\int _{0}^{x}e^{{x-t}}u(t)dt+u(x)\,

\cos(x)=\sin(x)+u(x)\,

u(x)=\cos(x)-\sin(x)\,


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