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.

Options

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

A

Explanation

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.