Final answer:
The solution to the given forced but undamped system with a specific frequency is a superposition of the general solution to the homogeneous equation and a particular solution. The general solution will be of the form u(t) = Ccos(t) + Dsin(t) + (3/ω2-1)cos(ωt), with the constants C and D determined by initial conditions.
Step-by-step explanation:
The student is dealing with a forced but undamped system that is described by a differential equation. Given the equation u'' + u = 3cos(ωt) with initial conditions u(0) = 0 and u'(0) = 0, we are asked to find the solution for u(t) when ω ≠ 1. In this case, the external force does not resonate with the system's natural frequency, therefore we expect the solution to involve a particular solution that represents the forced response plus a complementary solution that represents the natural motion.
The solution can be represented as u(t) = A cos(ωt + φ) plus a homogeneous solution A cos(t) + B sin(t), due to the characteristic equation r2+1=0. However, because ω ≠ 1, A and φ from the particular solution will not be the same as the coefficients from the homogeneous solution. To find the actual values of A and B, one would need to use the initial conditions to solve for them.
The general solution to the differential equation will look like u(t) = Ccos(t) + Dsin(t) + (3/ω2-1)cos(ωt), where C and D are constants determined by the initial conditions.