Write the solution as two cosine terms u\'\' + 49u = 5cos(4t), u(0) = 0, u\'(0)

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The characteristic equation is: r^2 + 49 = 0 the roots are: r = -7i , 7i so the homogeneous part of solution is: uh = Acos(7t) + Bsin(7t) To find particular solution we use undetermined coefficients method: p(t) = 5cos(4t) -> up = C.cos(4t) + Dsin(4t) up' = -4Csin(4t) + 4Dcos(4t) up" = -16Ccos(4t) - 16Dsin(4t) up" + 49up = -16Ccos(4t) - 16Dsin(4t) + 49Ccos(4t) + 49Dsin(4t) = 33Ccos(4t) + 33Dsin(4t) = 5cos(4t) C = 5/33 , D = 0 So the general solution is: u = uh + up = Acos(7t) + Bsin(7t) + (5/33)cos(4t) u(0) = A+5/33 = 0 -> A = -5/33 u'(0) = 7B = 20 -> B = 20/7 Therefore: u = (-5/33)cos(7t) + (20/7)sin(7t) + (5/33)cos(4t)

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