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--- book:chap4:4.5_linear_odes [2021/12/07 18:53] – abril
+++ book:chap4:4.5_linear_odes [2024/02/01 00:50] (current) – polish English jv
@@ Line 1: / Line 1: @@
-===== 4.5  Linear ODEs --- Particle suspended from a spring  =====
+===== 4.5  Linear ODEs — Particle suspended from a spring  =====
 There are two forces acting on a particle is suspended from a spring:
@@ Line 165: / Line 165: @@
 a pair of complex conjugated numbers, or
 we have to deal with the case $\gamma^2=4\,k/m$ where there only is a single root.
-We treat the cases one after the other.\\
+We treat the cases one after the other.
 **1.  Two real roots**\\
-In this case $\gamma^2 < 4\,k/m$ such that $\lambda_\pm \in \mathbb{R}_-$.
+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*}
   x(t) =  A_+ \; \mathrm{e}^{ \lambda_+ \, (t-t_0) } + A_- \; \mathrm{e}^{ \lambda_- \, (t-t_0) }
 \end{align*}
 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*}
   \left .