KVPY Sample Paper KVPY Stream-SX Model Paper-22

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

Correct Answer: A

Solution :

\[I.F.={{e}^{\int{-1\,dx}}}={{e}^{-x}}\]

\[\therefore \] \[y{{e}^{-x}}=\int{{{e}^{-x}}(\cos x-\sin x)}\,dx\]

\[=\int{{{e}^{-x}}((-1)\sin x+\cos x)}\,dx\]\[\Rightarrow \] \[{{e}^{-x}}\sin x+\operatorname{c}\]

\[\therefore \] \[y=\sin x+c\,\,{{e}^{x}}\]

Now \[\underset{x\,\to \,\infty }{\mathop{\lim }}\,{{e}^{x}}=\infty \]but since y is bounded.

\[\therefore \] \[c=0\]

\[\therefore \] \[y=\sin \] is the solution.