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Chapter 8: Problem 8

Prove that if \(x(t), y(t), t_{1}

Short Answer

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Question: Show that if \(x(t), y(t), t_{1}<t<t_{2}\) is a solution to an autonomous system of ordinary differential equations, then \(x(t+c), y(t+c)\) for any real constant \(c\) is also a solution. Show that this property does not hold for nonautonomous systems using the example \(\dot{x}=x, \dot{y}=tx\). Answer: The property of translation holds true for autonomous systems, which means that if given functions are solutions, then their translations by any constant will also be solutions. This property does not hold true for nonautonomous systems, as illustrated using the example \(\dot{x}=x, \dot{y}=tx\). In this case, the translated functions do not satisfy the nonautonomous system of equations.

Step by step solution

01

Substituting \(x(t+c)\) and \(y(t+c)\) into the differential equations

Given the system of differential equations \(\dot{x}=f(x, y),\ \dot{y}=g(x, y)\) and let's assume that the functions \(x(t+c)\) and \(y(t+c)\) are solutions to this system. We need to find the derivatives of these new functions with respect to \(t\). The derivative of \(x(t+c)\) with respect to \(t\) is \(\dot{x}(t+c)\). The derivative of \(y(t+c)\) with respect to \(t\) is \(\dot{y}(t+c)\). Now, we will substitute these functions into the given system of differential equations and check if they satisfy the equations.
02

Checking if the new functions satisfy the given system

Substitute the new functions into the differential equations: \(\dot{x}(t+c) = f(x(t+c), y(t+c))\) \(\dot{y}(t+c) = g(x(t+c), y(t+c))\) But we know that \(x(t)\) and \(y(t)\) satisfy the given system of differential equations. Therefore, \(\dot{x}(t) = f(x(t), y(t))\) \(\dot{y}(t) = g(x(t), y(t))\) Thus, by the property of translation, \(x(t+c)\) and \(y(t+c)\) also satisfy the given system of differential equations. So the property holds for autonomous systems. Now let's move on to the second part of the exercise to show that this property does not hold for nonautonomous systems.
03

Substituting \(x(t+c)\) and \(y(t+c)\) into the nonautonomous system

Consider the example given: \(\dot{x}=x, \dot{y}=tx\) Let's substitute \(x(t+c)\) and \(y(t+c)\) into these equations and check if they satisfy the equations: \(\dot{x}(t+c)=x(t+c)\) \(\dot{y}(t+c)=(t+c)x(t+c)\) Now, we will compare these equations with the original ones to find a contradiction.
04

Checking if the new functions satisfy the nonautonomous system

The original system of equations is: \(\dot{x}(t)=x(t)\) \(\dot{y}(t)=tx(t)\) Comparing the equations for \(x(t+c)\) and \(y(t+c)\) with the original ones, we can see that the first equation holds true, but the second equation does not. In the second equation, we have: \(\dot{y}(t+c)=(t+c)x(t+c)\) But in the original equation, we have: \(\dot{y}(t)=tx(t)\) These two equations are not the same, which means the property of translation does not hold for nonautonomous systems. So we have successfully shown that the property of translation holds for autonomous systems and does not hold for nonautonomous systems, as illustrated in the given example.

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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 equations that involve an unknown function and its derivatives. They play a pivotal role in modeling many natural and theoretical phenomena. Differential equations can be categorized mainly into two types:
  • Ordinary Differential Equations (ODEs): Involves derivatives of a function with respect to a single variable. For example, the differential equation \(\dot{x} = x\) is an ODE because it involves the derivative of \(x\) with respect to time \(t\).
  • Partial Differential Equations (PDEs): Involves derivatives with respect to more than one variable.
In the context of autonomous systems, the differential equations are expressed solely in terms of the dependent variables without explicit dependence on the independent variable, commonly time. This distinction is crucial because it affects the behavior and solutions of these equations. In contrast, equations that do involve the independent variable explicitly are known as nonautonomous systems.
Solution of System of Equations
The solution of a system of differential equations involves finding a set of functions that satisfy all equations within the system. The solution process often requires:
  • Initial Conditions: These are initial values given for the variables. They are necessary for determining a unique solution.
  • Analytical Methods: Such techniques involve solving the equations to get explicit solutions, using integration or other algebraic manipulations.
  • Numerical Methods: When analytical solutions are challenging or impossible, numerical methods like Euler's method or the Runge-Kutta method provide approximate solutions.
In autonomous systems, once a solution is found, any shift in time of this solution will also be a solution. This means if \(x(t)\) and \(y(t)\) are solutions, then \(x(t+c)\) and \(y(t+c)\) will also be solutions for any real constant \(c\). Contrast this with nonautonomous systems, where time-dependent factors, such as \(t\) in \(\dot{y} = tx\), make it such that time-shifted solutions do not automatically satisfy the system equations.
Nonautonomous Systems
Nonautonomous systems are systems in which the differential equations explicitly involve the independent variable (often time). This means an additional complexity layer where the system's behavior is directly influenced by time. In nonautonomous systems, properties like the time-translation property, which hold true for autonomous systems, no longer apply. For instance:
  • Time-Dependency: Here, the equations depend on a variable like time \(t\) directly, for example, \(\dot{y} = tx\).
  • Non-invariant Solutions: Solutions of nonautonomous systems are not necessarily invariant under time shifts, as demonstrated by the mismatch in solutions when shifting \(x(t+c)\) and \(y(t+c)\) into \(\dot{y} = t x\).
This characteristic exemplifies why solving nonautonomous systems requires careful attention to the timing of events described by the differential equations, posing unique challenges compared to autonomous systems.

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Most popular questions from this chapter

Determine the asymptotic behavior of the solution of each system near the critical point. Sketch the trajectories of the associated linear system. (a) \(\dot{x}=2 \sin x+y\), \(\dot{y}=\sin x-3 y .\) (b) \(\begin{aligned} & \dot{x}=-x-x^{2}+x y \\ & \dot{y}=-y+x y-y^{2} \\\ \text { (c) } & \dot{x}=x+e^{-y}-1, \\ & \dot{y}=-y-e^{-y}+1 \end{aligned}\)

Consider the Hamiltonian equation $$ \dot{q}=\frac{\partial H}{\partial p}, \quad \dot{p}=-\frac{\partial H}{\partial q} . $$ Let \(H(p, q)=T(p)+W(q)\), where \(T\) is the kinetic energy and \(W\) is the potential energy. If \(H\) is analytic in \(p\) and \(q\) and \(H(0,0)=0\), prove the Lagrange theorem: a position of stable equilibrium is a point at which the potential energy is a relative minimum.

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Consider the system $$ \begin{array}{l} \dot{x}=y-x f(x, y), \\ \dot{y}=-x-y f(x, y), \end{array} $$ where \(f(x, y)\) is analytic at the origin and \(f(0,0)=0\). Describe the relation between \(f(x, y)\) and the type of stability.

Describe the nature of the critical point of each system and sketch the trajectories. \(\begin{array}{lllll}\text { (a) } & \dot{x}=x, & \text { (b) } & \dot{x}=-x+2 y, & \text { (c) } & \dot{x}=2 x-8 y, \\ & \dot{y}=2 x+2 y . & & \dot{y}=x-y . & & \dot{y}=x-2 y . \\ \text { (d) } & \dot{x}=-x, & \text { (e) } & \dot{x}=-x+y, & \text { (f) } & \dot{x}=-3 x+2 y \text { , } \\ & \dot{y}=x-y . & \dot{y}=2 x . & \dot{y}=-2 x .\end{array}\)
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