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\textbf{Math 2552 -- Differential Equations} & & \textbf{Worksheet 11} (Feb 18, ch4 intro) \\
Georgia Institute of Technology, Spring 2019 & & Intro to second order equations \\
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\textit{Note:}\quad As chapter 4 has not yet been covered in lecture, this is merely an intro to get you thinking about the topic.
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\item Solve the second order ODE by first transforming into a system of first order ODE:
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\item $6y''-y'-y=0$
\item $4y''-9y=0$
\item $y''+6y'+16y=0$
\item $y''+3y'+2y=0,\quad y(0)=2,\ y'(0)=-1$
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\item (\textit{optional})\quad Introduction to the theory of second order ODE:
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\item (\textit{review})\quad What can you say about the existence and uniqueness of a solution to the following IVP? $$\frac{d\mathbf x}{dt} = \begin{pmatrix}p_{11}(t) & p_{12}(t)\\p_{21}(t) & p_{22}(t)\end{pmatrix}\mathbf x + \begin{pmatrix}g_1(t)\\g_2(t)\end{pmatrix},\quad \mathbf x(t_0)=\mathbf x_0$$
\item Write the following IVP as a system of first order equations: $$y''+p(t)y'+q(t)y=g(t),\quad y(t_0)=y_0,\ y'(t_0)=y_1$$
\item Based on your answers for a and b, what do you think is the equivalent existence and uniqueness theorem for the equation in part b?
\item After transforming into a system of first order ODE, what is the characteristic equation for $ay''+by'+cy=0$? Do the roots (i.e., the eigenvalues) change if you multiply the characteristic equation by $a$?
\item Suppose you have the equation $ay''+by'+cy=0$ as in part d, and that the eigenvalues are distinct, real valued. Find a good choice of eigenvectors, and provide a general solution.
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