/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 \(\dot{x}=1-2 \cos x\)... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 6

\(\dot{x}=1-2 \cos x\)

Short Answer

Expert verified
The function \(x(t)\) is not possible to be found analytically due to the non-elementary integral, \(\int \frac{dx}{1-2 \cos x}\). However, an approximate solution can be found by separating the variables, performing the integration, and representing the antiderivative of the left side of the equation as \(F(x)\), so we can express \(x(t)\) as \(F^{-1}(t + C)\), where \(F^{-1}\) is the inverse function of \(F(x)\) and \(C\) is a constant.

Step by step solution

01

Separate the variables

To find x(t), we first need to separate the variables on each side of the equation, so that x terms are on one side and t terms are on the other: \(\frac{dx}{dt} = 1-2 \cos x \) Now, we separate variables by multiplying dt and dividing by the (1 - 2cos(x)): \(\frac{dx}{1 - 2\cos x}=dt\).
02

Integration of both sides

Now integrate both sides with respect to their respective variables: \(\int \frac{dx}{1-2 \cos x} = \int dt\). As mentioned before, the left-hand side of the equation is not an elementary integral, so an exact result cannot be found. Instead, we can use numerical integration methods to find an approximate solution.
03

Evaluate the integrals

First, we will evaluate the right side of the equation, where we integrate \(dt\): \(\int dt = t + C_1\), where \(C_1\) is a constant of integration. Next, suppose we represent the antiderivative of the left side of the equation as \(F(x)\). Since we cannot find an exact expression, we can write it as: \(\int \frac{dx}{1-2 \cos x} = F(x) + C_2\), where \(C_2\) is another constant of integration.
04

Solve for x(t)

Now, we can equate both sides of the equation: \(F(x) + C_2 = t + C_1\) Or, by rearranging: \(F(x) = t + C\), where \(C = C_1 - C_2\) is constant. So, we have: \(x(t) = F^{-1}(t + C)\), where \(F^{-1}\) is the inverse function of \(F(x)\). Note that we still cannot find an exact solution for x(t) due to the non-elementary integral on the left-hand side. However, this equation shows how to find an approximate solution for x(t) using inverse functions and a constant of integration.

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Key Concepts

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

Differential Equations
Differential equations are mathematical tools used to describe many phenomena in science and engineering, ranging from motion in physics to population growth in biology. At its core, a differential equation relates a function to its derivatives, expressing how the function changes over time or space. The equation given in the exercise, \(\dot{x}=1-2 \cos x\), is a first-order differential equation, where the dot over the x represents the first derivative of x with respect to time, t.

In the context of nonlinear dynamics, such equations often describe systems with complex behavior where the relationship between the system's state and its rate of change is not proportional. Solving a differential equation means finding the function x(t) that satisfies the equation for a given condition. While some differential equations have closed-form solutions, others, like the one in our exercise, may require numerical methods or approximations to solve.
Separation of Variables
Separation of variables is a technique commonly used to solve ordinary differential equations (ODEs) when it is possible to separate the variables involved into two sides of an equation. This means rearranging the terms so that each variable, along with its differential, is on one side of the equation on its own. In this exercise, we did so by manipulating the original equation, resulting in \(\frac{dx}{1 - 2\cos x}=dt\).

This method significantly simplifies the complexity of solving ODEs as it allows us to focus on each variable independently. After separation, we proceed to solve the equation by integrating both sides. This approach hinges on the fundamental theorem of calculus, which links integration and differentiation as inverse processes. However, when the integral cannot be expressed in terms of elementary functions, as seen in the left-hand side of original equation, we must seek alternative methods, such as numerical integration, to continue.
Numerical Integration
Numerical integration, also known as numerical quadrature, is a key method in computational mathematics used to evaluate integrals when an analytical solution is difficult or impossible to find. Instead of seeking an exact closed-form solution, numerical integration approximates the value of an integral using various algorithms, which break the domain into small segments and estimate the area under the curve.

Common methods include the trapezoidal rule and Simpson's rule, among others. In our exercise, numerical integration would allow us to approximate \(F(x)\), the antiderivative of \(\frac{dx}{1-2 \cos x}\), since an exact expression cannot be determined. With the estimated \(F(x)\), we can then find an approximate solution for the function \(x(t)\) by including the constants of integration. This process highlights that even when faced with non-elementary integrals, we are not without recourse; numerical methods extend our capability to solve complex differential equations in nonlinear dynamics.

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Most popular questions from this chapter

(Error estimate for Euler method) In this question you'll use Taylor series expansions to estimate the error in taking one step by the Euler method. The exact solution and the Euler approximation both start at \(x=x_{0}\) when \(t=t_{0}\). We want to compare the exact value \(x\left(t_{1}\right)=x\left(t_{0}+\Delta t\right)\) with the Euler approximation \(x_{1}=x_{0}+f\left(x_{0}\right) \Delta t\) a) Expand \(x\left(t_{1}\right) \equiv x\left(t_{0}+\Delta t\right)\) as a Taylor series in \(\Delta t\), through terms of \(O\left(\Delta t^{2}\right)\). Express your answer solely in terms of \(x_{0}, \Delta t\), and \(f\) and its derivatives at \(x_{e}\). b) Show that the local error \(\left|x\left(t_{l}\right)-x_{1}\right| \sim C(\Delta t)^{2}\) and give an explicit expression for the constant \(C\). (Generally one is more interested in the global error incurred after integrating over a time interval of fixed length \(T=n \Delta t .\) Since each step produces an \(O(\Delta t)^{2}\) error, and we take \(n=T / \Delta t=O\left(\Delta t^{-1}\right)\) steps, the global error \(\left|x\left(t_{k}\right)-x_{n}\right|\) is \(O(\Delta t)\), as claimed in the text)

\(\dot{x}=1+\frac{1}{2} \cos x\)

Analyze the following equations graphically. In each case, sketch the vector field on the real line, find all the fixed points, classify their stability, and sketch the graph of \(x(t)\) for different initial conditions. Then try for a few minutes to obtain the analytical solution for \(x(t)\); if you get stuck, don't try for too long since in several cases it's impossible to solve the equation in closed form! \(\dot{x}=4 x^{2}-16\)

(Critical slowing down) In statistical mechanics, the phenomenon of "critical slowing down" is a signature of a second-order phase transition. At the transition, the system relaxes to equilibrium much more slowly than usual. Here's a mathematical version of the effect: a) Obtain the analytical solution to \(\dot{x}=-x^{3}\) for an arbitrary initial condition. Show that \(x(t) \rightarrow 0\) as \(t \rightarrow \infty\), but that the decay is not exponential. (You should find that the decay is a much slower algebraic function of \(t\).) b) To get some intuition about the slowness of the decay, make a numerically accurate plot of the solution for the initial condition \(x_{0}=10\), for \(0 \leq t \leq 10\). Then, on the same graph, plot the solution to \(\dot{x}=-x\) for the same initial condition.

(A general example of non-uniqueness) Consider the initial value problem \(\dot{x}=|x|^{p / q}, x(0)=0\), where \(p\) and \(q\) are positive integers with no common factors. a) Show that there are an infinite number of solutions if \(pq\).
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