Use variation of parameters to find a particular solution, given the solutions y1,y2\displaystyle {y}_{{1}},{y}_{{2}}y1,y2 of the complementary equation
sin(x)y′′+(2sin(x)−cos(x))y′+(sin(x)−cos(x))y=e−x\displaystyle {\sin{{\left({x}\right)}}}{y}{''}+{\left({2}{\sin{{\left({x}\right)}}}-{\cos{{\left({x}\right)}}}\right)}{y}'+{\left({\sin{{\left({x}\right)}}}-{\cos{{\left({x}\right)}}}\right)}{y}={e}^{{-{x}}}sin(x)y′′+(2sin(x)−cos(x))y′+(sin(x)−cos(x))y=e−x
y1=e−x,y2=e−xcos(x)\displaystyle {y}_{{1}}={e}^{{-{x}}},\quad{y}_{{2}}={e}^{{-{x}}}{\cos{{\left({x}\right)}}}y1=e−x,y2=e−xcos(x)
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