Verify that y=sin xcos x-cos x is a solution of the initial value problem y'+(tan x)y=cos^2 xy(0)=-1 on the interval -pi /2 < x< pi/2

Caylee Villegas 2022-07-27 Answered
Verify that y = sin x cos x cos x is a solution of the initial value problem y + ( tan x ) y = c o s 2 x y ( 0 ) = 1 on the interval π / 2 < x < π / 2
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Answers (1)

slapadabassyc
Answered 2022-07-28 Author has 21 answers
y'+y P(x)= Q(x)
y + ( tan x ) y = cos 2 x linear diff. eqn.
P ( x ) = tan x
Q ( x ) = cos 2 x
integration factor i ( x ) = exp ( P ( x ) d x ) = exp ( tan ( x ) d x )
= exp ( ln ( cos x ) = 1 cos x
y . i ( x ) = i ( x ) Q ( x ) d x + c
y 1 cos x = 1 cos x cos 2 x d x + c
y 1 cos x = sin x + c
y = sin x . cos x + c . cos x
y ( 0 ) = 1 1 = sin 0 cos 0 + c . c o s 0 c = 1
y = sin x . cos x cos x
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