Verify that `y=A\ cos x+sinx`satisfies the differential equation `cosx \ (dy)/(dx)+(sinx) \ y=1.`

Explanation

Given: `cosx (dy)/(dx)+(sinx) y=1` ………..(1)


Given: `y=A cos x+sin x`


Differentiating the equation


`frac{d y}{d x}=-A sin x+cos x`


Put `frac{d y}{d x}=-A sin x+cos x and y=A cos x+sin x` in equation(1),


LHS `=(cos x)(-A sin x+cos x)+(sin x)(A cos x+sin x) `


`=-A sin x cos x+cos ^{2} x+A cos x sin x+sin ^{2} x`


`=cos ^{2} x+sin ^{2} x`


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