Question

If \(a, b,\) and \(c\) are positive constants, show that all solutions of \(a y^{\prime \prime}+b y^{\prime}+c y=0\) approach zero as \(t \rightarrow \infty\).

Short answer

Question: Prove that all solutions of the second-order homogeneous linear differential equation \(ay''+by'+cy=0\) approach zero as \(t \rightarrow \infty\) whenever \(a\), \(b\), and \(c\) are positive constants. Answer: To prove that all solutions approach zero as \(t \rightarrow \infty\), we first solve the given differential equation by finding the general solution for each case depending on the roots of the characteristic equation. In all three possible cases (real and distinct roots, real and equal roots, complex conjugate roots), the exponential parts of the general solutions approach zero as \(t\) approaches infinity, given that \(a\), \(b\), and \(c\) are positive constants. This shows that all solutions of the given differential equation approach zero as \(t \rightarrow \infty\).

Step-by-step solution

Write the characteristic equation

To solve the second-order homogeneous linear differential equation, we first write the characteristic equation, which is given by: \(r^2a+ rb + c = 0\)

Calculate roots of the characteristic equation

Now, to find the roots of the characteristic equation, we simply apply the quadratic formula: \(r = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\) There are three different cases for the roots based on the determinant (\(b^2-4ac\)): real and distinct roots, real and equal roots, or complex conjugate roots.

Write down the solution for each case

For each case, we write down the general solution of the given differential equation as follows: 1. If the roots are real and distinct: \(r_1\) and \(r_2\), then the general solution of the given differential equation is \(y(t) = C_1 e^{r_1 t} + C_2 e^{r_2 t}\) 2. If the roots are real and equal: \(r_1 = r_2 = r\), then the general solution of the given differential equation is \(y(t) = C_1 e^{r t} + C_2 t e^{r t}\) 3. If the roots are complex conjugates: \(r_1 = \alpha + \beta i\) and \(r_2 = \alpha - \beta i\), then the general solution of the given differential equation is \(y(t) = e^{\alpha t} (C_1 \cos(\beta t) + C_2 \sin(\beta t))\)

Analyze the behavior of the solutions as \(t \rightarrow \infty\)

In order to prove that the solutions approach zero as \(t \rightarrow \infty\), we need to show that the exponential part of the solution goes to zero as \(t\) approaches infinity: 1. For the case of real and distinct roots, both roots must be negative, since \(a, b,\) and \(c\) are positive constants. Thus, \(e^{r_1 t}\) and \(e^{r_2 t}\) both approach zero as \(t \rightarrow \infty\). 2. For the case of real and equal roots, the root must again be negative. Thus, both \(e^{r t}\) and \(t e^{r t}\) approach zero as \(t \rightarrow \infty\). 3. For the case of complex conjugate roots, the real part (\(\alpha\)) of the root must be negative in order for the solution to approach zero as \(t \rightarrow \infty\). Since the exponential term is \(e^{\alpha t}\), it approaches zero when the real part of the root is negative, thus making the whole solution approach zero as \(t \rightarrow \infty\). As we can see, in all three cases, the solutions approach zero as \(t \rightarrow \infty\), proving the given statement.

Key concepts

Characteristic Equation

In the context of second-order homogeneous linear differential equations, the characteristic equation plays a critical role in finding the general solution. Consider the generic form of such a differential equation:
\[\begin{equation}a y^{\text{\text{prime} \text{prime}} }+b y^{\text{prime} }+c y=0\text{,}\end{equation}\] where a, b, and c are coefficients, often constants, and y is a function of some variable, typically time or space. The characteristic equation associated with this differential equation is a quadratic equation obtained by assuming a solution of the form y(t) = e^{rt}. Substituting this assumed solution into the differential equation yields the characteristic equation:
\[\begin{equation}r^2a + rb + c = 0\text{.}\end{equation}\] The solutions r to this quadratic equation, known as the roots, determine the behavior of the original differential equation's solutions. They can be found using the quadratic formula:
\[\begin{equation}r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\text{.}\end{equation}\] The nature of these roots—whether they are real and distinct, real and equal, or complex conjugate pairs—is crucial, as each scenario leads to a different general solution for y(t).

Real and Distinct Roots

Within the study of homogeneous linear differential equations, encountering real and distinct roots implies an exponential nature in the solutions.
When applying the quadratic formula to the characteristic equation and finding that the discriminant (b^2-4ac) is positive, two distinct real roots r_1 and r_2 emerge. The general solution for the differential equation when the roots are real and distinct is given by:
\[\begin{equation}y(t) = C_1 e^{r_1 t} + C_2 e^{r_2 t}\text{,}\end{equation}\] where C_1 and C_2 are constants determined by initial conditions or boundary values. The solution exhibits two separate exponential terms, each corresponding to one of the roots of the characteristic equation. In practice, should these roots be negative—which they typically are when constants a, b, and c in the differential equation are positive—the solution demonstrates a decay toward zero as t approaches infinity, highlighting a stable system behavior.

Complex Conjugate Roots

When dealing with differential equations, complex conjugate roots indicate oscillatory behavior in the system's response. These roots occur if the discriminant of the characteristic equation (b^2-4ac) is negative, leading to a pair of roots r_1 = \alpha + \beta i and r_2 = \alpha - \beta i, where i is the imaginary unit, and \alpha and \beta are real numbers. The general solution takes on a distinct form in contrast to the case with real roots:
\[\begin{equation}y(t) = e^{\alpha t} (C_1 \cos(\beta t) + C_2 \sin(\beta t))\text{,}\end{equation}\] where e^{\alpha t} represents the exponential growth or decay depending on the sign of \alpha, and (C_1 \cos(\beta t) + C_2 \sin(\beta t)) reflects the oscillatory component of the solution. If \alpha is negative—which is the case when a, b, and c are positive—the oscillations diminish over time, resulting in the solution approaching zero as t goes to infinity. This synthesis of growth and oscillation often emerges in systems with damping, where the presence of complex roots signals a return to equilibrium in a non-monotonic fashion.