I have the following problem:
A $8\,\mathrm{kg}$ mass stretches a spring $1.96\,\mathrm{m}$. At $t=0$, an external force $f(t)=2\cos(2t)$ is applied to the system. The damping constant is $3\,\mathrm{N}\cdot\mathrm{s}/\mathrm{m}$. Find its position $u(t)$.
I know how to solve the ODE, but my question is: may I assume $u(0)=u'(0)=0$? If so, why? Or does the general solution depends on the constants?
On
Since the system is damped, initial conditions only affect the motion of the mass for a transient period of time, for that reason it seems reasonable assume rest conditions as a starting point.
For the given data we have $\omega_0=2.24 \;rad/s$ and $\zeta =8.4\;\%$, assuming rest initial condition this is the graph for $u(t)$:
For free oscillation with initial conditions $u(0)=0.20 \;m$ and $u'(0)=0 \;m/s$ this is the graph for u(t):
The most complete answer would give names to the initial values, say $u_0 := u(0)$ and $v_0 := u'(0)$, then provide a solution $u(t)$ that explicitly depends on those two quantities.
Then, if you want to, you could make assumptions about their values. For me, the most intuitive read of the problem statement is that $u_0 = 1.96$ and $v_0 = 0$ (i.e., the mass begins motionless at its most stretched position), which differs from the initial values that you suggested.