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%Here's a basic document. I recommend to use it as your starting document for all your latex projects. Just make copies of it, and modify them. Also, add your favorite packages and structure as your latex learning curve progresses.
\title{title}
\author{Name\\Department}
\date{\today}
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\paragraph{Problem 1a.} The equation $\ddot{x}+0.2\dot{x}+2x=2\cos t$ is a non-homogeneous equation of order two. Consequently, its analytic solution is $\psi(t)=\psi_p(t)+\psi_h(t)$. The characteristic polynomial being $\lambda^2+0.2\lambda+2=0$, we find the roots
\[ \lambda_1=-0.1+i\sqrt{1.99},\qquad \lambda_2=-0.1-i\sqrt{1.99}. \]
As a result, we have
\[\psi_h(t)= c_1e^{\lambda_1 t}+c_2e^{\lambda_2 t}.\]
Now, since the roots are complex conjugates of the form $\alpha\pm i\beta$, and since $e^{iz}=\cos z+i\sin z$ and $e^{-iz}=\cos z-i\sin z$, we have awesome stuff.
\end{document}