KVPY Sample Paper KVPY Stream-SX Model Paper-8

Solution of the differential equation \[\frac{dy}{dx}-y=\cos x-\sin x\] satisfying the condition that y should be bounded when

A) \[y=\sin x\]

B) \[y=\cos x\]

C) \[y=\sin x+\cos x\]

D) None of these

Correct Answer: A

Solution :

I.F. = \[I.F.={{e}^{\int{-Idx}}}={{e}^{-x}}\]

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

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

\[={{e}^{-x}}\sin x+c\]

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

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

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

\[\therefore \,\,\,y=\sin x\]is the solution