Problem 16
Question
A model of a spring/mass system is \(4 x^{\prime \prime}+i x=0\). By inspection of the differential equation only, discuss the behavior of the system over a long period of time.
Step-by-Step Solution
Verified Answer
The system will oscillate indefinitely without damping due to the imaginary roots with no damping.
1Step 1: Identify the Type of Differential Equation
The given differential equation is \(4x'' + ix = 0\). We need to identify it as an ordinary linear differential equation with constant coefficients.
2Step 2: Write the Characteristic Equation
Converting the differential equation into a characteristic equation, we assume a solution of the form \(x = e^{kt}\), leading to \(4k^2 + i = 0\).
3Step 3: Solve the Characteristic Equation
Solving \(4k^2 + i = 0\), we find \(k^2 = -\frac{i}{4}\), resulting in the complex roots \(k = \pm \frac{1}{2}i^{1/2}\).
4Step 4: Analyze the Roots for System Behavior
The imaginary roots indicate oscillatory behavior. Since they don't include a real component, the system will oscillate indefinitely without damping over time.
5Step 5: Discuss the Impact of the Imaginary Component
The presence of the imaginary number \(i\) in the equation means the system is governed by complex forces, but without a damping term (real part), these forces drive perpetual oscillations.
Key Concepts
Oscillatory BehaviorOrdinary Differential EquationsComplex Coefficients
Oscillatory Behavior
In a spring-mass system like the one given by the equation \(4x'' + ix = 0\), the oscillatory behavior is crucial to understand. Oscillations occur naturally in systems where there is a restoring force that acts to bring the system back to its equilibrium position.
For the given system, after solving the characteristic equation, it is clear that we have purely imaginary roots. This result is significant because it tells us there is no real component in the solutions. The roots are \(k = \pm \frac{1}{2} i^{1/2}\), indicating that the system will not experience any decay or growth in amplitude over time.
The absence of a damping term means that the system will oscillate indefinitely. This behavior is typical in idealized systems where energy loss is not a factor. The oscillations will maintain a constant frequency determined by the imaginary component of the roots, resulting in smooth, consistent cycles.
For the given system, after solving the characteristic equation, it is clear that we have purely imaginary roots. This result is significant because it tells us there is no real component in the solutions. The roots are \(k = \pm \frac{1}{2} i^{1/2}\), indicating that the system will not experience any decay or growth in amplitude over time.
The absence of a damping term means that the system will oscillate indefinitely. This behavior is typical in idealized systems where energy loss is not a factor. The oscillations will maintain a constant frequency determined by the imaginary component of the roots, resulting in smooth, consistent cycles.
Ordinary Differential Equations
Ordinary differential equations (ODEs) are equations involving functions and their derivatives. They are used to model various phenomena such as motion, growth, decay, and rest. In this context, ODEs describe how the position of a mass on a spring changes over time.
The particular ODE provided, \(4x'' + ix = 0\), is a second-order linear ODE with constant coefficients. The term ordinary indicates the equation depends on a single independent variable, which in most cases is time. Here, both the mass and spring constants determine the form of the ODE, reflecting how the system evolves.
Solving an ODE like this one involves finding the function \(x(t)\) that satisfies the equation. This process typically involves converting the problem into a simpler form, such as a characteristic equation, to analyze the behavior of the solution.
The particular ODE provided, \(4x'' + ix = 0\), is a second-order linear ODE with constant coefficients. The term ordinary indicates the equation depends on a single independent variable, which in most cases is time. Here, both the mass and spring constants determine the form of the ODE, reflecting how the system evolves.
Solving an ODE like this one involves finding the function \(x(t)\) that satisfies the equation. This process typically involves converting the problem into a simpler form, such as a characteristic equation, to analyze the behavior of the solution.
Complex Coefficients
Complex coefficients in differential equations add an additional layer of complexity to the system's behavior. In the given problem, the term \(ix\) indicates the presence of a complex coefficient.
When converting to the characteristic equation, this results in complex roots. Solving the equation \(4k^2 + i = 0\) reveals these roots are purely imaginary.
The impact of complex coefficients is primarily seen in the oscillatory nature and stability of the system. Since there is no real component in the solution, the presence of complex coefficients alone will cause pure oscillations without decay. Thus, complex coefficients in an ODE like this imply that forces causing oscillation do not diminish over time, resulting in a perpetual, undamped system behavior.
Understanding these effects helps in predicting and analyzing physical systems where complex phenomena govern behaviors.
When converting to the characteristic equation, this results in complex roots. Solving the equation \(4k^2 + i = 0\) reveals these roots are purely imaginary.
The impact of complex coefficients is primarily seen in the oscillatory nature and stability of the system. Since there is no real component in the solution, the presence of complex coefficients alone will cause pure oscillations without decay. Thus, complex coefficients in an ODE like this imply that forces causing oscillation do not diminish over time, resulting in a perpetual, undamped system behavior.
Understanding these effects helps in predicting and analyzing physical systems where complex phenomena govern behaviors.
Other exercises in this chapter
Problem 16
In Problems \(1-20\), solve the given system of differential equations by systematic elimination. $$ \begin{aligned} D^{2} x-2\left(D^{2}+D\right) y &=\sin t \\
View solution Problem 16
In Problems 11-20, find the eigenvalues and eigenfunctions for the given boundary-value problem. $$ y^{\prime \prime}+\lambda y=0, y(-\pi)=0, y(\pi)=0 $$
View solution Problem 16
In Problems 1-18, solve the given differential equation. $$ x^{3} y^{\prime \prime \prime}+x y^{\prime}-y=0 $$
View solution Problem 16
In Problems \(1-18\), solve each differential equation by variation of parameters. $$ 2 y^{\prime \prime}+2 y^{\prime}+y=4 \sqrt{x} $$
View solution