Each graph in Figure 11.102 represents a solution to one of...

Each graph in Figure 11.102 represents a solution to one of the differential equations: (a) $x^{\prime \prime}+x=0$ (b) $x^{\prime \prime}+4 x=0$ (c) $x^{\prime \prime}+16 x=0$ Assuming the $t$ -scales on the four graphs are the same, which graph represents a solution to which equation? Find an equation for each graph.

Each graph in Figure 11.102 represents a solution to one of the differential equations:
(a) $x^{\prime \prime}+x=0$
(b) $x^{\prime \prime}+4 x=0$
(c) $x^{\prime \prime}+16 x=0$
Assuming the $t$ -scales on the four graphs are the same, which graph represents a solution to which equation? Find an equation for each graph.

Key Concepts

Angular Frequency and Period

Angular frequency (?) is a key parameter that influences the period of the oscillation, where the period T is given by T = 2?/?. Different values of ? in the differential equation lead to solutions that oscillate with different speeds and thus have different periods; a higher ? results in faster oscillations with a shorter period.

Graphical Interpretation of Oscillatory Behavior

Interpreting the graphs of these solutions involves comparing the observed periods of oscillation to the theoretical relationship T = 2?/?. By matching the period measured from a graph with this formula, one can determine which ? (and hence which differential equation) corresponds to a given graph, allowing for identification of the specific harmonic oscillator represented.

Characteristic Equation and Imaginary Roots

The method of solving these equations involves forming the characteristic equation r² + ?² = 0, which yields complex conjugate roots r = ±i?. These imaginary roots lead to solutions expressed in terms of sine and cosine functions, indicating that the system exhibits periodic or oscillatory behavior.

Linear Homogeneous Second-Order Differential Equations

These differential equations have the form x'' + ?²x = 0, where the equation is homogeneous and the order is two, meaning that any solution is a linear combination of two independent solutions. The behavior of the solutions is determined entirely by the constant coefficient associated with x, which directly influences the dynamics and oscillatory nature of the system.

Simple Harmonic Motion

The differential equation x'' + ?²x = 0 commonly models simple harmonic motion, a fundamental concept in oscillatory systems such as springs and pendulums. The general solution, combining sine and cosine functions, describes how the displacement of the system evolves with time in a predictable, sinusoidal pattern.

Transcript

Please give Ace some feedback