Question

Without finding their characteristic roots, determine whether the following differential equations will give rise to convergent time paths: (a) \(y^{\prime \prime \prime}(t)-10 y^{\prime \prime}(t)+27 y^{\prime}(t)-18 y=3\) (b) \(y^{\prime \prime \prime}(t)-11 y^{\prime \prime}(t)+34 y^{\prime}(t)+24 y=5\) (c) \(y^{\prime \prime \prime}(t)+4 y^{\prime \prime}(t)-5 y^{\prime}(t)-2 y=-2\)

Short answer

Equations (a), (b), and (c) potentially lead to divergent time paths.

Step-by-step solution

Understand the nature of the differential equations

The provided differential equations are third-order linear differential equations with constant coefficients. Such equations can be written in the standard form: \( y'''(t) + ay''(t) + by'(t) + cy = f(t) \). The convergence of solutions depends on the roots of its characteristic equation being negative or having negative real parts.

Write the characteristic equation

The characteristic equation is created by replacing \( y(t) \) with \( e^{ ext{rt}} \) in the homogeneous part of the differential equation. For \( n^{th} \) order differential equation in standard form, it is \( r^3 + ar^2 + br + c = 0 \). This will be used to analyze the stability without finding the specific roots.

Verify the convergence conditions

For the time path to be convergent, all the real parts of the roots of the characteristic equation must be negative. Since we are not calculating the exact roots, we check the coefficients: summing up all the coefficients \( a, b, \) and \( c \). If these sums show specific conditions (like based on stability criteria where \( a, b, c > 0 \)), it suggests roots could be negative.

Analyze equation (a)

The characteristic equation for (a) is \( r^3 - 10r^2 + 27r - 18 = 0 \). The signs of the coefficients are: \( 1, -10, 27, -18 \). Since the characteristic polynomial includes a sign change, it is indicative that at least one real positive root might be present.

Analyze equation (b)

The characteristic equation for (b) is \( r^3 - 11r^2 + 34r + 24 = 0 \). The signs of the coefficients are: \( 1, -11, 34, 24 \). A positive leading coefficient and a positive constant with no sign change in all except \( r^3 \) indicates a non-negative real root could exist, suggesting possible divergence.

Analyze equation (c)

The characteristic equation for (c) is \( r^3 + 4r^2 - 5r - 2 = 0 \). The signs of the coefficients are: \( 1, 4, -5, -2 \). A negative term (\(-5r\)) with an increasing order of positive coefficients suggests an unstable configuration leading to potential divergence.

Summarize findings

Based on the sign and arrangement of coefficients, equations (a), (b), and (c) may have positive real root components. This implies that all given differential equations potentially have divergent solutions.

Key concepts

Convergent Time Paths

When studying differential equations, it's crucial to understand the concept of convergent time paths. This concept is particularly important when dealing with solutions that depend on time, such as those deriving from differential equations involving growth or decay. A convergent time path indicates that the solution of a differential equation approaches a stable value as time progresses. For a differential equation to have a convergent solution, the roots of its characteristic equation must have negative real parts. This means that as time goes to infinity, the solution diminishes and stabilizes. It's akin to predicting that a slowly rippling pond will eventually become still, with surface waves disappearing over time. However, determining convergence without solving for these roots requires insight into the equation's structure and coefficients. The inspection technique involves assessing the sign and stability of the roots indirectly by analyzing these coefficients.

Characteristic Equation

The characteristic equation is a key component in solving linear differential equations. It helps us analyze the behavior of solutions, especially in terms of convergence or divergence. The characteristic equation is essentially a polynomial equation derived from the differential equation by substituting derivatives with terms of exponential functions.For an nth-order linear differential equation with constant coefficients, the characteristic equation is obtained by replacing each derivative by powers of a variable, usually denoted as 'r'. It takes the form:\[ r^n + ar^{n-1} + br^{n-2} + \ldots + c = 0 \]This substitution transforms the differential equation into an algebraic polynomial. Solving this polynomial for 'r' helps us determine the nature of the differential equation's solutions. Specifically, the roots of this characteristic equation dictate the response of the system, whether it converges, remains steady, or diverges over time.

Roots of Polynomial

The nature of the roots of a polynomial greatly influences the behavior of its corresponding differential equation. For a third-order differential equation, the characteristic polynomial is a cubic equation, with roots that may be real or complex. The roots can provide insights into several aspects:

• If all roots are real and negative, the solution converges, indicating stability.
• Complex roots with negative real parts also suggest convergence.
• Any positive real root will lead to exponential growth, indicating diverging behavior.

In many real-world scenarios, simply identifying these roots helps predict the long-term behavior of the system. For instance, chemical reactions that stabilize over time or mechanical systems that dampen due to friction can often be modeled using this principle.

Stability of Solutions

Stability refers to how a system responds to small disturbances or changes over time. In the context of differential equations, stability is about whether solutions head towards a fixed state or not. For a system to be stable, the solutions of its differential equations must tend to a steady state when subjected to negligible disturbances. The sign of the real parts of the roots of the characteristic equation plays a decisive role in determining this stability:

• All negative real roots indicate that solutions decay to zero, showing stability.
• If any root has a positive real part, the solution will grow unbounded, illustrating instability and divergence.

Understanding stability is crucial in fields like engineering and control systems, where predicting and ensuring the desired performance of systems over time is essential. Stability assessments prevent undesirable outcomes, such as uncontrolled oscillations or runaway reactions.