roots of D(2D+4)y=0
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Answer:
What is the CF of a differential equation (D^4+1) y=0?
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It is a homogeneous linear differential equation of IV order with constant coefficients and so is easy to solve. The corresponding auxiliary equation is m^4 + 1 = 0, whose roots are the four complex 4th roots of (-1) = cos(pi) + i sin(pi). These are
cos(pi/4) (+/-) i sin (pi/4) = (1/sqrt 2) +/- i/sqrt 2 and
cos (3.pi/4) (+/-) i sin(3.pi/4) = (-1/sqrt 2) +/- i/sqrt 2. .
Accordingly a general solution (i.e. Complementary Function) is
y = e^(x/sqrt 2)[A cos (x/sqrt 2) + B sin (x/sqrt 2)] +
e^(-x/sqrt/2)[C cos(x/sqrt 2) + D sin(x/sqrt 2)].
A detailed treatment may be found in ‘Ordinary Differential Equations - An Introduction’, by B.Rai & others (Narosa Publishing House, New Delhi)