Problem 15 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Refer to the equation $$x^{\prime \prime}+b x^{\prime}+c x=0 .$$ Suppose \(b<0\) and \(c>0 .\) For \(x(0)\) and \(x^{\prime}(0)\) not both zero, is it possible that \(\lim _{t \rightarrow \infty}|x(t)|=L\), where \(L\) is finite? Explain.

Step-by-Step Solution

Answer

No, it is not possible for the limit of |x(t)| as t approaches infinity to be a finite value L other than zero.

Step 1: Analyzing the differential equation

The given equation is a second order linear homogeneous differential equation. The general solution to this equation can be found by finding the roots of the characteristic equation, \[m^2+bm+c=0.\] By using the quadratic formula, the roots become \[m_{1,2}=(-b \pm \sqrt{b^2 - 4c}) / 2.\] Depending on the discriminant \(b^2 - 4c\), there are three cases: two real distinct roots if \(b^2 - 4c > 0\), two identical real roots if \(b^2 - 4c = 0\), or complex roots if \(b^2 - 4c < 0\).

Step 2: Determining the type of roots

The exercise mentions \(b < 0\) and \(c > 0\). This tells us that the discriminant \(b^2 - 4c\) will always be negative, because b squared (a positive value) is less than 4 times c (also a positive value), implying that the roots of the characteristic equation will be complex.

Step 3: Evaluating the general solution

In the case of complex roots, the general solution of the differential equation is \[x(t) = e^{rt}(A\cos(\omega t) + B\sin(\omega t)).\] Here, \(r\) and \(\omega\) are respectively the real and the imaginary part of the complex roots. We know that \(r = -b/2 < 0\) from the calculation of the roots. Thus, the exponential part will decay as t approaches to infinity.

Step 4: Conclude the solution

Since \(r = -b/2 < 0\), as \(t \rightarrow \infty\), \(e^{rt} \rightarrow 0\), while the \((A\cos(\omega t) + B\sin(\omega t))\) part oscillates between -1 and 1. Therefore, \[ \lim_{t \rightarrow \infty}|x(t)|=0 \] regardless of the values of initial conditions \(x(0)\) and \(x'(0)\) that are not both zero. It is thus concluded that it is not possible for the limit of |x(t)| as t approaches infinity to be a finite value L other than 0.

Key Concepts

Characteristic Equation

When working with second order linear homogeneous differential equations, a key concept is the characteristic equation. For the equation \[ x'' + bx' + c x = 0, \] the characteristic equation is derived by assuming a solution of the form \( x(t) = e^{mt} \). Substituting into the differential equation results in a quadratic equation in \( m \):\[ m^2 + bm + c = 0. \]Solving this characteristic equation is crucial since the roots, \( m_1 \) and \( m_2 \), determine the behavior of the solutions of the differential equation.

• If \( b^2 - 4c > 0 \), we get two distinct real roots.
• If \( b^2 - 4c = 0 \), we get a repeated real root.
• If \( b^2 - 4c < 0 \), we encounter complex roots.

These roots are pivotal in shaping the form of the general solution to the differential equation. Understanding how to form and solve the characteristic equation is a foundational skill in solving these equations.

Complex Roots

In the context of linear differential equations, when the discriminant \( b^2 - 4c \) is negative, the roots of the characteristic equation are complex. This happens under certain conditions of the coefficients, as seen in the exercise where \( b < 0 \) and \( c > 0 \), making \( b^2 - 4c \) always negative.The complex roots will be in the form:\[ m_1, m_2 = (-b \pm i \sqrt{4c - b^2}) / 2. \]This means each root has a real part, \( r = -b/2 \), and an imaginary part, \( \omega = \sqrt{4c - b^2}/2 \). The general solution involving complex roots is:\[ x(t) = e^{rt}(A\cos(\omega t) + B\sin(\omega t)), \]where:

• \( e^{rt} \) is an exponential term that controls the amplitude of the solution.
• \(\cos(\omega t)\) and \(\sin(\omega t)\) represent the oscillatory nature of the solution due to the imaginary part \( \omega \).

Since \( r < 0 \), \( e^{rt} \) approaches zero as \( t \) tends to infinity, leading the entire solution towards zero, illustrating the decaying oscillation.

Homogeneous Differential Equation

A homogeneous differential equation is one where all terms are made up of the function \( x(t) \) and its derivatives, and there are no independent terms. In this problem,\[ x'' + bx' + c x = 0, \]is such a homogeneous differential equation. It is called homogeneous because the right-hand side is zero. This equation means that out of all solutions, we are particularly interested in those whose behavior depends solely on the relationship and interaction between \( x(t) \), its derivatives, and the coefficients in the equation.Solving a homogeneous differential equation involves finding the characteristic equation and solving for its roots, leading to the general solution:\[ x(t) = e^{rt}(A\cos(\omega t) + B\sin(\omega t)), \]as seen from the exercise, especially when roots are complex.Key characteristics of homogeneous equations include:

• The solution set forms a vector space.
• Superposition principle applies — sums of solutions are also solutions.

Understanding these allows us to find the full spectrum of solutions that satisfy both the differential equation and given initial conditions.