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Chapter 11: Problem 36

Transform the second-order differential equation $$ \frac{d^{2} x}{d t^{2}}+\frac{d x}{d t}=2 x $$ into a system of first-order differential equations.

Short Answer

Expert verified
The transformed system is: \(\frac{dy_1}{dt} = y_2\), \(\frac{dy_2}{dt} = 2y_1 - y_2\).

Step by step solution

01

Identify Variables and Equations

Given the second-order differential equation \(\frac{d^{2} x}{d t^{2}}+\frac{d x}{d t}=2 x\), we identify the variable \(x\) dependent on time \(t\). The goal is to express this equation as a system of first-order differential equations.
02

Introduce a Substitution

Define a new variable \(y_1 = x\) and use this definition to facilitate reducing the order of the differential equation. This will allow us to express higher derivatives in terms of first-order derivatives.
03

Define New Variables for Derivatives

Introduce a second variable \(y_2 = \frac{dx}{dt}\). This substitution helps express the second derivative \(\frac{d^2x}{dt^2}\) in terms of first-order derivatives with respect to \(y_1\) and \(y_2\).
04

Express the System of Equations

From \(y_1 = x\), we have \(\frac{dy_1}{dt} = y_2\). The second-order differential equation \(\frac{d^{2} x}{d t^{2}}+\frac{d x}{d t}=2 x\) can be rewritten using \(y_1\) and \(y_2\) as \(\frac{dy_2}{dt} = 2y_1 - y_2\).
05

System Formulation

We have now transformed the original second-order differential equation into the following system of first-order equations: \(\frac{dy_1}{dt} = y_2\) and \(\frac{dy_2}{dt} = 2y_1 - y_2\). This is the required transformation of equations.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

First-Order Differential Equations
First-order differential equations are equations that involve the first derivative of a function but no higher derivatives. They play a fundamental role in differential equations because they can describe a wide range of phenomena, from growth processes to decay processes, motion, and more.
Here's a clearer understanding:
  • In a first-order differential equation, the highest derivative is the first derivative, typically written as \( \frac{dy}{dt} \), \( \frac{dx}{dt} \), or similar.
  • The objective with these equations is often to find a function that satisfies the equation, which involves one of its derivatives.
  • These equations can be linear or nonlinear depending on the function's terms.

In the context of the given problem, we transformed a second-order equation into two first-order differential equations. This is a common practice because first-order systems are often easier to analyze and solve.
Second-Order Differential Equations
Second-order differential equations involve the second derivative of a function, and they frequently appear in physics and engineering, particularly those involving acceleration or forces. Let's dive deeper into their nature:
  • In these equations, you often see expressions like \( \frac{d^2y}{dt^2} \), which denotes the second derivative of \( y \) with respect to \( t \).
  • They can describe systems with memory or inertia, such as the oscillation of a spring or an electric circuit.
  • These equations may include terms of the first derivative and the function itself, adding to their complexity.

The provided equation \( \frac{d^{2} x}{d t^{2}} + \frac{d x}{d t} = 2x \) is a second-order linear differential equation. By transforming it into a system of first-order equations, we simplify the process of finding solutions.
System of Equations
A system of equations consists of two or more equations with multiple variables that are solved together. When handling differential equations, converting to a first-order system can simplify the analysis and computational solutions. Consider these important points about systems of equations:
  • A system of first-order equations is essentially a set of equations containing first derivatives, often used to model dynamic systems in various fields.
  • Each equation in the system is interlinked with others, allowing us to understand complex relationships between variables.
  • The method of solving often involves techniques such as substitution, elimination, or matrix operations like determining eigenvalues and eigenvectors in linear algebra.

In the exercise, we took the initial second-order differential equation and created a system of equations: \( \frac{dy_1}{dt} = y_2 \) and \( \frac{dy_2}{dt} = 2y_1 - y_2 \). This system captures the original equation's behavior using first-order dynamics, a valuable technique for solving complex differential equations in various practical applications.

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Most popular questions from this chapter

We consider differential equations of the form \(\frac{d x}{d t}=A x(t)\) where \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right]\) Analyze the stability of the equilibrium \(\mathbf{( 0 , 0 ) , \text { and classify the }}\) equilibrium. \(A=\left[\begin{array}{rr}-2 & 3 \\ 1 & -4\end{array}\right]\)

Use the eigenvalue approach to analyze all equilibria of the given Lotka- Volterra models of interspecific competition. \(\frac{d N_{1}}{d t}=3 N_{1}\left(1-\frac{N_{1}}{18}-1.3 \frac{N_{2}}{18}\right)\) \(\frac{d N_{2}}{d t}=2 N_{2}\left(1-\frac{N_{2}}{20}-0.6 \frac{N_{1}}{20}\right)\)

We consider differential equations of the form \(\frac{d \mathbf{x}}{d t}=A \mathbf{x}(t)\) where $$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] $$ The eigenvalues of A will be real, distinct, and nonzero. Analyze the stability of the equilibrium \((0,0)\), and classify the equilib\mathrm{\\{} r i u m ~ a c c o r d i n g ~ t o ~ w h e t h e r ~ i t ~ i s ~ a ~ s i n k , ~ a ~ s o u r c e , ~ o r ~ a ~ s a d d l e ~ point. \(A=\left[\begin{array}{rr}-1 & 3 \\ 2 & 4\end{array}\right]\)

Biological Control Agent Assume that \(N(t)\) denotes the density of an insect species at time \(t\) and \(P(t)\) denotes the density of its predator at time \(t\). The insect species is an agricultural pest, and its predator is used as a biological control agent. Their dynamics are given by the system of differential equations $$ \begin{array}{l} \frac{d N}{d t}=5 N-3 P N \\ \frac{d P}{d t}=2 P N-P \end{array} $$ (a) Explain why $$ \frac{d N}{d t}=5 N $$ describes the dynamics of the insect in the absence of the predator. Solve (11.75). Describe what happens to the insect population in the absence of the predator. (b) Explain why introducing the insect predator into the system can help to control the density of the insect. (c) Assume that at the beginning of the growing season the insect density is \(0.5\) and the predator density is 2. You decide to control the insects by using an insecticide in addition to the predator. You are careful and choose an insecticide that does not harm the predator. After you spray, the insect density drops to \(0.01\) and the predator density remains at \(2 .\) Use a graphing calculator to investigate the long-term implications of your decision to spray the field. In particular, investigate what would have happened to the insect densities if you had decided not to spray the field, and compare your results with the insect density over time that results from your application of the insecticide.

We consider differential equations of the form \(\frac{d \mathbf{x}}{d t}=A \mathbf{x}(t)\) where $$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] $$ The eigenvalues of A will be real, distinct, and nonzero. Analyze the stability of the equilibrium \((0,0)\), and classify the equilib\mathrm{\\{} r i u m ~ a c c o r d i n g ~ t o ~ w h e t h e r ~ i t ~ i s ~ a ~ s i n k , ~ a ~ s o u r c e , ~ o r ~ a ~ s a d d l e ~ point. \(A=\left[\begin{array}{rr}1 & 3 \\ 1 & -1\end{array}\right]\)
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