book:chap4:4.5_linear_odes
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| book:chap4:4.5_linear_odes [2024/02/01 00:48] – fix typo jv | book:chap4:4.5_linear_odes [2024/02/01 00:50] (current) – polish English jv | ||
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| **1. Two real roots**\\ | **1. Two real roots**\\ | ||
| - | In this case $\gamma^2 > 4\, | + | In this case $\gamma^2 > 4\,k/m$, and $\lambda_\pm \in \mathbb{R}_-$. |
| - | The motion of the oscillator is described by | + | Hence, the motion of the oscillator is described by |
| \begin{align*} | \begin{align*} | ||
| x(t) = A_+ \; \mathrm{e}^{ \lambda_+ \, (t-t_0) } + A_- \; \mathrm{e}^{ \lambda_- \, (t-t_0) } | x(t) = A_+ \; \mathrm{e}^{ \lambda_+ \, (t-t_0) } + A_- \; \mathrm{e}^{ \lambda_- \, (t-t_0) } | ||
| \end{align*} | \end{align*} | ||
| which is a real-valued function for amplitudes $A_\pm \in \mathbb{R}$. | which is a real-valued function for amplitudes $A_\pm \in \mathbb{R}$. | ||
| - | The solution for the initial conditions $x(t_0) = x_0$ and $\dot x(t_0)=v_0$ | + | Moreover, the solution for the initial conditions $x(t_0) = x_0$ and $\dot x(t_0)=v_0$ |
| - | is then found by solving the equations | + | is found by solving the equations |
| \begin{align*} | \begin{align*} | ||
| \left . | \left . | ||
book/chap4/4.5_linear_odes.1706744919.txt.gz · Last modified: 2024/02/01 00:48 by jv