(Solved): 1. A critically damped oscillator of unit mass with natural frequency is subjected to an ext ...

1. A critically damped oscillator of unit mass with natural frequency $?$ is subjected to an external force $f(t)=f_0{e}^{??t}$, where $?$ is a positive constant and $??=?$. The equation of motion has the form $\ddot{x}+2?\stackrel{?}{x}+{?}^{2}x=f(t).$ (a) Determine a particular solution $x_p(t)$ to the above equation. (b) Write down the general solution $x_h(t)$ to the corresponding homogeneous differential equation. mark] $[1$ (c) Find the complete solution $x(t)$ using the initial conditions $x(0)=0,\stackrel{?}{x}(0)=0$. $[1$ mark] (d) Find the limiting form of the solution in (c) when $???$. Your answer should involve only $f_0,?$ and $t$. [1 mark]