Question

find the solution of the given initial value problem. Sketch the graph of the solution and describe its behavior for increasing\(t.\) $$ y^{\prime \prime}-2 y^{\prime}+5 y=0, \quad y(\pi / 2)=0, \quad y^{\prime}(\pi / 2)=2 $$

Short answer

Based on the step-by-step solution, the particular solution for the given initial value problem is: $$ y(t) = e^t[e^{-\pi / 2} \cos(2t) - e^{-\pi / 2} \sin(2t)] $$ As $t$ increases, the graph of this particular solution will have an increasing oscillatory behavior with an exponentially increasing amplitude.

Step-by-step solution

Solve the differential equation

First, we will write down the given second-order linear homogeneous ODE with constant coefficients: $$ y^{\prime\prime} - 2y^{\prime} + 5y = 0 $$ Next, we will establish the characteristic equation for this ODE: $$ r^2 - 2r + 5 = 0 $$

Find the roots of the characteristic equation

To find the roots of the characteristic equation, we will use the quadratic formula: $$ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ Plugging in the coefficients of the characteristic equation gives us: $$ r = \frac{2 \pm \sqrt{(-2)^2 - 4(1)(5)}}{2(1)} = 1 \pm 2i $$ Since we have complex roots, we can write the general solution to the ODE in the form: $$ y(t) = e^{1t}(A \cos(2t) + B \sin(2t)) $$ where \(A\) and \(B\) are constants to be determined using the initial conditions.

Apply the initial conditions

We are given the initial conditions \(y(\pi / 2) = 0\) and \(y^{\prime}(\pi / 2) = 2\). We'll first find the first derivative of the general solution: $$ y^{\prime}(t) = e^{t}[(-2A \sin(2t) + 2B \cos(2t)) + (A \cos(2t) + B \sin(2t))] $$ Now we'll use the initial conditions to find the values of \(A\) and \(B\). Initial condition 1: \(y(\pi / 2) = 0\) $$ 0 = e^{\pi / 2}(A \cos(\pi) + B \sin(\pi)) $$ Since \(\cos(\pi) = -1\) and \(\sin(\pi) = 0\), we have \(A = e^{-\pi / 2}\). Initial condition 2: \(y^{\prime}(\pi / 2) = 2\) $$ 2 = e^{\pi / 2}[(2e^{-\pi / 2} \cos(\pi) + 2B \sin(\pi)) + (e^{-\pi / 2} \cos(\pi) + B \sin(\pi))] $$ Since \(\cos(\pi) = -1\) and \(\sin(\pi) = 0\), we have \(B = -e^{-\pi / 2}\).

Write down the particular solution

Using the initial conditions, we have found that \(A = e^{-\pi / 2}\) and \(B = -e^{-\pi / 2}\). Substituting these values into the general solution, we obtain the particular solution for the given IVP: $$ y(t) = e^t[e^{-\pi / 2} \cos(2t) - e^{-\pi / 2} \sin(2t)] $$

Sketch the graph and describe the behavior

To sketch the graph and determine the behavior for increasing \(t\), we have to analyze the components of our solution. As t increases, the term \(e^{t}\) will increase exponentially. The sinusoidal functions, \(\cos(2t)\) and \(\sin(2t)\), are oscillatory functions with amplitudes bounded between -1 and 1. Therefore, as \(t\) increases, the graph of the solution will have an increasing oscillatory behavior with an exponentially increasing amplitude.

Key concepts

Second-order Differential Equations

Second-order differential equations are equations that involve the unknown function, its first derivative, and its second derivative. In the form of \[ y'' + p(t)y' + q(t)y = g(t) \] in these equations, \(y''\) represents the second derivative of \(y\), while \(p\), \(q\), and \(g\) are functions of \(t\).

These equations are prevalent in modeling real-world problems since they can describe dynamic systems, such as mechanical vibrations and electrical circuits. In our exercise, we encounter a linear homogeneous second-order differential equation with constant coefficients, expressed as: \[ y^{rac{}{}''} - 2y' + 5y = 0 \] This formulation is useful since these types of equations have well-defined solution methods, often involving the characteristic equation.

Complex Roots of Characteristic Equation

To solve second-order differential equations, we often rely on their characteristic equations, which transform the differential equation into an algebraic one. The characteristic equation for our example is: \[ r^2 - 2r + 5 = 0 \] Using the quadratic formula, we find the roots of this equation: \[ r = 1 \pm 2i \] Since these roots are complex, we can express them in terms of real and imaginary components. Here, 1 is the real part, and \(2i\) is the imaginary part.

The general solution to the differential equation when the characteristic roots are complex is:

• \( e^{ ext{real part } t}(A \cos( ext{imaginary part } t) + B \sin( ext{imaginary part } t)) \)

In our problem: \[ y(t) = e^{1t} (A \cos(2t) + B \sin(2t)) \]

Complex roots indicate oscillatory behavior in the solution, which is characterized by sine and cosine functions.

Exponential Growth in Solutions

The solution we derive from our differential equation often experiences exponential growth, particularly noticeable in the term \( e^t \). This exponential behavior implies that as time \( t \) increases, the part of the solution represented by \( e^t \) also grows without bound.

In our problem:

• The term \( e^t \) multiplies the oscillatory components \((A \cos(2t) + B \sin(2t))\), indicating that any oscillations will expand in amplitude over time.

This aspect of the solution is critical for understanding systems that might, for instance, depict an escalating amplitude in a physical scenario.
Exponential growth suggests that small changes early on might lead to large variations at later times, influencing predictions about system behaviors.

Oscillatory Behavior in Differential Equations

Oscillatory behavior in a differential equation's solution indicates that the output of the system will fluctuate or swing back and forth regularly over time. This is typically embodied by sine (\( \sin \)) and cosine (\( \cos \)) functions within the solution.

In our exercise, the terms \( \cos(2t) \) and \( \sin(2t) \) indicate such oscillations. This leads to a situation where:

• The solution oscillates with a frequency influenced by the imaginary part of the complex roots, which in this case is 2.
• The amplitude of the oscillations, however, is determined by the coefficients \(A\) and \(B\).

The solution graph would show wave-like behavior, but because it includes an exponential term \( e^t \), the oscillations' amplitude grows larger over time. This is a compound behavior where exponential increase and oscillations interplay. This kind of behavior often appears in systems where forces get repeated or cycling over time, such as in circuits or pendulums.