Question

Consider the initial value problem $$ u^{\prime \prime}+\gamma u^{\prime}+u=0, \quad u(0)=2, \quad u^{\prime}(0)=0 $$ We wish to explore how long a time interval is required for the solution to become "negligible" and how this interval depends on the damping coefficient \(\gamma\). To be more precise, let us seek the time \(\tau\) such that \(|u(t)|<0.01\) for all \(t>\tau .\) Note that critical damping for this problem occurs for \(\gamma=2\) (a) Let \(\gamma=0.25\) and determine \(\tau,\) or at least estimate it fairly accurately from a plot of the solution. (b) Repeat part (a) for several other values of \(\gamma\) in the interval \(0<\gamma<1.5 .\) Note that \(\tau\) steadily decreases as \(\gamma\) increases for \(\gamma\) in this range. (c) Obtain a graph of \(\tau\) versus \(\gamma\) by plotting the pairs of values found in parts (a) and (b). Is the graph a smooth curve? (d) Repeat part (b) for values of \(\gamma\) between 1.5 and \(2 .\) Show that \(\tau\) continues to decrease until \(\gamma\) reaches a certain critical value \(\gamma_{0}\), after which \(\tau\) increases. Find \(\gamma_{0}\) and the corresponding minimum value of \(\tau\) to two decimal places. (e) Another way to proceed is to write the solution of the initial value problem in the form (26). Neglect the cosine factor and consider only the exponential factor and the amplitude \(R\). Then find an expression for \(\tau\) as a function of \(\gamma\). Compare the approximate results obtained in this way with the values determined in parts (a), (b), and (d).

Short answer

**Question:** For the initial value problem \(u^{\prime \prime}+\gamma u^{\prime}+u=0\) with \(u(0)=2\) and \(u^{\prime}(0)=0\), estimate the time \(\tau\) when \(|u(t)|<0.01\), following the given steps. **Answer:** To find the time \(\tau\) when \(|u(t)|<0.01\) for the given initial value problem, follow the steps below: 1. Solve the second-order linear ODE to find the general solution. 2. Estimate \(\tau\) for \(\gamma=0.25\). 3. Repeat step 2 for several other values of \(\gamma\) within the range \(0<\gamma<1.5\). 4. Plot \(\tau\) vs \(\gamma\) and observe the graph. 5. Repeat step 2 for values of \(\gamma\) between \(1.5\) and \(2\). 6. Find an expression for \(\tau\) as a function of \(\gamma\). By following these steps, you can estimate the time \(\tau\) when \(|u(t)|<0.01\) for the given initial value problem.

Step-by-step solution

Solve the second-order linear ODE

First, we need to find the general solution of the given second-order linear ODE: $$ u^{\prime \prime}+\gamma u^{\prime}+u=0 $$ This is a homogenous ODE with constant coefficients. To find the general solution, we'll use the characteristic equation corresponding to this ODE: $$ r^2 + \gamma r + 1 = 0 $$ We can solve this quadratic equation and find the roots \(r_1\) and \(r_2\). The roots will determine the general solution of the ODE: 1. If \(r_1\) and \(r_2\) are real and distinct, the general solution is $$ u(t) = C_1e^{r_1t}+C_2e^{r_2t} $$ 2. If \(r_1\) and \(r_2\) are complex conjugates, say \(r_1=a+bi\) and \(r_2=a-bi\), the general solution is $$ u(t) = e^{at}[C_1\cos(bt) + C_2 \sin(bt)] $$ 3. If \(r_1\) and \(r_2\) are equal (both real), say \(r_1=r_2=a\), the general solution is $$ u(t) = (C_1 + C_2t)e^{at} $$ Now, we'll find the specific solution that satisfies the initial conditions \(u(0)=2\) and \(u^{\prime}(0)=0\).

Estimate \(\tau\) for \(\gamma=0.25\)

For this step, we will plug in the given value \(\gamma = 0.25\) into the quadratic equation: $$ r^2 + 0.25r + 1 = 0 $$ This equation has complex conjugate roots. Therefore, we can find the specific solution using the second case mentioned in Step 1. Then, we can plot the solution and find the time \(\tau\) such that \(|u(t)|<0.01\) for all \(t>\tau\).

Repeat step 2 for several other values of \(\gamma\) within the range \(0

For this step, we need to compute \(\tau\) for different values of \(\gamma\) in the range \(0<\gamma<1.5\). We can follow the same procedure as in Step 2 and compare the behavior of \(\tau\) as \(\gamma\) increases.

Plot \(\tau\) vs \(\gamma\) and observe the graph

Now that we have determined \(\tau\) for different values of \(\gamma\), we can plot the pairs of values \((\tau, \gamma)\) to visualize their relationship. We will analyze the shape of the graph and determine if it is a smooth curve.

Repeat step 2 for values of \(\gamma\) between \(1.5\) and \(2\)

In this step, we need to compute \(\tau\) for various values of \(\gamma\) between \(1.5\) and \(2\) following the same procedure as in Step 2. This will allow us to find the critical value \(\gamma_{0}\) where the behavior of \(\tau\) changes. Lastly, we will calculate the minimum value of \(\tau\) corresponding to \(\gamma_{0}\) up to two decimal places.

Find an expression for \(\tau\) as a function of \(\gamma\)

Using the given form in the exercise, we will focus on the exponential factor and the amplitude \(R\). We will then derive an expression for \(\tau\) based on \(\gamma\), forming a function \(\tau(\gamma)\). Finally, we will compare the approximated results of \(\tau(\gamma)\) to the exact values determined in steps 2, 3, 4, and 5.

Key concepts

Damping coefficient

In the realm of differential equations, the damping coefficient \( \gamma \) plays a vital role in predicting the behavior of solutions. It represents the force that opposes the motion of an object, acting like a braking mechanism that slows down oscillations.
When studying mechanical systems, the damping coefficient helps us understand how fast the movement stops over time. In mathematical terms, it forms part of the equation \( u'' + \gamma u' + u = 0 \), where it influences the roots of the characteristic equation, ultimately affecting the system's behavior.

• Low damping (e.g., \( \gamma < 1 \)): leads to a slow reduction in the amplitude of oscillation, giving rise to oscillatory solutions.
• Higher damping (e.g., \( \gamma \) close to critical damping \( = 2 \)): results in a faster decay, leading to a smooth curve without oscillations.

Thus, understanding the damping coefficient is crucial for analyzing a system's long-term behavior.

Characteristic equation

The characteristic equation is a critical gateway to solving linear differential equations. It helps us determine the nature of solutions without actually solving the differential equation fully. For our equation \( u'' + \gamma u' + u = 0 \), the characteristic equation simplifies to \( r^2 + \gamma r + 1 = 0 \). This is a quadratic equation in the variable \( r \), representing the possible 'rates' at which the solution decays or grows.
There are three possible scenarios:

• Two distinct real roots: leading to solutions that consist of exponential decays.
• Repeated real roots: resulting in a critically damped system with a different form of exponential decay.
• Complex conjugate roots: resulting in oscillatory behavior modulated by an exponential decay.

This equation essentially tells us everything about how solutions will behave over time, providing us with insights even before evaluating specific initial conditions.

Initial value problem

Initial value problems (IVPs) are a foundational concept in differential equations, where the goal is to find a unique solution that satisfies given starting conditions. In the context of our differential equation \( u'' + \gamma u' + u = 0 \), we start with the initial conditions \( u(0) = 2 \) and \( u'(0) = 0 \).
These initial conditions play a critical role in determining the specific path of the solution alongside the characteristic equation. They allow us to calculate the constants present in the general solution \( C_1 \) and \( C_2 \), thereby tailoring the mathematical solution to our real-world scenario.
By imposing these conditions, we focus on a specific solution path out of potentially infinite possibilities, making the IVP an indispensable tool for providing meaningful answers to real-world problems.

Exponential decay

Exponential decay is a fundamental concept that arises in many natural processes. In terms of differential equations, it describes how the amplitude of a function decreases over time. When solving the equation \( u'' + \gamma u' + u = 0 \), exponential decay often characterizes the behavior of \( u(t) \), especially for systems with damping.
The solution typically includes terms of the form \( e^{at} \), where \( a \) is a negative value leading to decay. This type of solution shows how the magnitude of the solution diminishes, eventually becoming negligible. The speed of this decay is directly influenced by the damping coefficient \( \gamma \) and the specific roots of the characteristic equation.
Exponential decay plays a vital role in predicting how quickly a system settles into a stable state, thus serving as a pivotal concept in understanding dynamic systems.

Complex conjugates

Complex conjugates emerge in the solutions of quadratic equations, notably in those cases where the discriminant is negative, implying the presence of oscillatory components within real systems. In the characteristic equation \( r^2 + \gamma r + 1 = 0 \), complex roots appear when the plot indicates underdamped behavior.
When roots are complex conjugates, they take the form \( a \pm bi \), with the solution expressing itself as \( e^{at}(C_1 \cos(bt) + C_2 \sin(bt)) \). This result indicates oscillations modulated by exponential decay, a scenario common in mechanical and electrical systems.
Thus, complex conjugates are crucial in modeling real-world phenomenons where periodic motion with gradual decay is observed, providing depth to how differential equations benefit from their interplay with complex numbers.