Let a solution `y=y(x)` of the differential equation `(dy)/(dx)cosx+y sin x-1` satisfy y(0)=1 Statement-1: `y(x)=sin((pi)/(4)+x)` Statement-2: The integrating factor of the given differential equation is sec x.A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1.B. Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-1.C. Statement-1 is True, Statement-2 is False.D. Statement-1 is False, Statement-2 is True.

Let a solution `y=y(x)` of the differential equation `(dy)/(dx)cosx+y

1.	Let a solution `y=y(x)` of the differential equation `(dy)/(dx)cosx+y sin x-1` satisfy y(0)=1 Statement-1: `y(x)=sin((pi)/(4)+x)` Statement-2: The integrating factor of the given differential equation is sec x.A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1.B. Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-1.C. Statement-1 is True, Statement-2 is False.D. Statement-1 is False, Statement-2 is True.
Answer» Correct Answer - A We have, `(dy)/(dx)cos x+y sin x=1` `rArr" "(dy)/(dx)+y tan x=sec x` This is a linear differential equation with I.F. given by `"I.F."=e^(inttanxdx)=e^(logsecx)=secx` So, statement-2 is true. Multiplying both sides of by I.F. = sec x and integrating w.r. to x, we get `y sec x=tan x+C" ...(ii)"` It is given that y = 1 when x = 0. `therefore" 1 = C"` Putting C = 1 in (ii), we get `ysec x =tan x+1` `rArr" "y=sin x +cos x=sqrt2 sin ((pi)/(4)+x)` So, statement-1 is true and statement-2 is a correct explanation for statement-1.

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