Question

The P.I. of (D^2+1) y=cos⁡〖3x 〗is *


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Answer 1

Answer:

The auxiliary equation is (m^2 +1)^2 = 0.

The roots are +/- i , +/- i.

The imaginary roots are repeated.

Hence the Complementary Function (CF) is (c1+c2 x)cos x + (c3+c4 x)sin x.

The Particular Integral (PI) = [1/f(D)] cos 3x = [1/(D^2 +1)^2] cos 3x

Applying the [1/f(D^2)] cos ax = [1/f(-a^2)]cos ax

we get [1/64] cos 3x

Hence the complete solution is = CF +PI

= (c1+c2 x)cos x + (c3+c4 x)sin x + [1/64] cos 3x