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Answer 1 · 2016-07-21T08:45:47

#y = 2(cos x - sin x) #

#(d^2y)/(dx^2) + y = 0#

characteristic equation: #r^2 + 1 = 0, r^2 = -1, = pm i# leads to general solution

#y = A e^(ix) + B e^(-ix)#

which is better expressed using Euler's formula as
#y = A cos x + B sin x#

applying IV's

#2 = A cos 0 + B sin 0 implies A = 2#

#-2 = 2 cos (pi/2) + B sin (pi/2) implies B = -2#

so

#y = 2(cos x - sin x) #