(Solved): \[ a \ddot{x}+b \dot{x}+c x=F(t) \] involves the following steps. 1. Write \( F(t) \) as the real ...

\[ a \ddot{x}+b \dot{x}+c x=F(t) \] involves the following steps. 1. Write \( F(t) \) as the real or imaginary part of a complex expression \( G(t)=Q(t) e^{(\alpha+i \beta) t} \), where \( Q(t) \) is a polynomial of degree \( m \). 2. Find a complex particular solution \( z_{p}(t) \) of the complex differential equation \[ a \ddot{z}+b \dot{z}+c z=G(t) \] by defining \[ z_{p}(t)=t^{s} e^{(\alpha+i \beta) t}\left(C_{m} t^{m}+C_{m-1} t^{m-1}+\cdots+C_{1} t+C_{0}\right) \] and solving for the complex coefficients \( C_{0}, C_{1}, \ldots, C_{m} \). Here \( s=0 \) if \( \alpha+i \beta \) is not a root of the characteristic equation. \( s=1 \) if \( \alpha+i \beta \) is a simple root of the characteristic equation. 3. Define the real solution \( x_{p}(t) \) by letting \[ x_{p}(t)=\operatorname{Re}\left(z_{p}(t)\right) \quad \text { or } \quad x_{p}(t)=\operatorname{Im}\left(z_{p}(t)\right), \] depending on whether \( F(t)=\operatorname{Re}(G(t)) \) or \( F(t)=\operatorname{Im}(G(t)) \). For the differential equation \[ \ddot{x}+4 x=2 t^{3} \sin (2 t) . \] 1. Find \( G(t)=Q(t) e^{(\alpha+i \beta) t} \) such that \( 2 t^{3} \sin (2 t)=\operatorname{Re}(G(t)) \) or \( 2 t^{3} \sin (2 t)=\operatorname{Im}(G(t)) \) \[ Q(t)= \] \[ \alpha+i \beta= \] 2. Using C0,C1,C2,C3,... etc. for \( C_{0}, C_{1}, C_{2}, C_{3}, \ldots \), enter the appropriate form of the particular complex solution \( z_{p}(t) \) \[ \begin{array}{l|l} z_{p}(t)= & e^{(\alpha+i \beta) t} \end{array} \] 3. How is the real particular solution \( x_{p}(t) \) obtained from the complex solution \( z_{p}(t) \) ? \[ x_{p}(t)=\left(z_{p}(t)\right) \] Enter Re for taking the real part or \( \operatorname{Im} \) for taking the imaginary part.