• # question_answer Solution of the differential equation $\frac{dy}{dx}\,-\,y\,=\,\cos \,\,x\,-\,\sin \,x$ satisfying the condition that y should be bounded when $x\,\to \,+\,\infty$is A) $y\,=\,\sin \,x$ B) $y\,=\,\,\cos \,x$ C) $y\,=\,\,\sin \,x\,+\,\cos \,x$ D) none of these

 $I.F.={{e}^{\int{-1\,dx}}}={{e}^{-x}}$ $\therefore$      $y{{e}^{-x}}=\int{{{e}^{-x}}(\cos x-\sin x)}\,dx$ $=\int{{{e}^{-x}}((-1)\sin x+\cos x)}\,dx$$\Rightarrow$            ${{e}^{-x}}\sin x+\operatorname{c}$ $\therefore$      $y=\sin x+c\,\,{{e}^{x}}$ Now $\underset{x\,\to \,\infty }{\mathop{\lim }}\,{{e}^{x}}=\infty$but since y is bounded. $\therefore$      $c=0$ $\therefore$      $y=\sin$ is the solution.