Solve-for-u-and-v-in-the-system-of-equations-below-u-e-2x-cos2x-v-e-2x-sin2x-0-u-e-2x-cos2x-v-e-2x-sin2x-xe-2x-sinx-where-u-and-v-are-functions-of-x-

Question Number 108180 by Ar Brandon last updated on 15/Aug/20

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{u}\:\mathrm{and}\:\mathrm{v}\:\mathrm{in}\:\mathrm{the}\:\mathrm{system}\:\mathrm{of}\:\mathrm{equations}\:\mathrm{below} \\ $$$$\begin{cases}{\mathrm{u}'\left(\mathrm{e}^{−\mathrm{2x}} \mathrm{cos2x}\right)+\mathrm{v}'\left(\mathrm{e}^{−\mathrm{2x}} \mathrm{sin2x}\right)=\mathrm{0}}\\{\mathrm{u}'\left(\mathrm{e}^{−\mathrm{2x}} \mathrm{cos2x}\right)'+\mathrm{v}'\left(\mathrm{e}^{−\mathrm{2x}} \mathrm{sin2x}\right)'=\mathrm{xe}^{−\mathrm{2x}} \mathrm{sinx}}\end{cases} \\ $$$$\mathrm{where}\:\mathrm{u}\:\mathrm{and}\:\mathrm{v}\:\mathrm{are}\:\mathrm{functions}\:\mathrm{of}\:\mathrm{x} \\ $$

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