Question

Given that \(x=\phi(t), y=\psi(t)\) is a solution of the autonomous system $$ d x / d t=F(x, y), \quad d y / d t=G(x, y) $$ for \(\alpha

Short answer

Question: Show that if x = φ(t) and y = ψ(t) are solutions for the autonomous system, \[ \frac{dx}{dt} = F(x, y) \] \[ \frac{dy}{dt} = G(x, y) \] for the range \(\alpha < t < \beta\), then x = Φ(t) = φ(t - s) and y = Ψ(t) = ψ(t - s) are also solutions for the autonomous system for the range \(\alpha + s < t < \beta + s\) for any real number s.

Step-by-step solution

Differentiate Φ(t) and Ψ(t) with respect to t

Now, we will find the derivatives \(\frac{dΦ}{dt}\) and \(\frac{dΨ}{dt}\) using the chain rule. \[ \frac{dΦ}{dt} = \frac{d\phi(t-s)}{dt} = \phi'(t-s)\frac{d(t-s)}{dt} = \phi'(t-s) \] \[ \frac{dΨ}{dt} = \frac{d\psi(t-s)}{dt} = \psi'(t-s)\frac{d(t-s)}{dt} = \psi'(t-s) \]

Rewrite F and G in terms of Φ and Ψ

Using the initial information, we can write the following equations: \[ \frac{dx}{dt} = F(x, y) \Rightarrow \frac{d\phi}{dt} = F(\phi(t), \psi(t)) \] \[ \frac{dy}{dt} = G(x, y) \Rightarrow \frac{d\psi}{dt} = G(\phi(t), \psi(t)) \] Now we need to rewrite the functions F and G in terms of Φ(t) and Ψ(t): \[ F(x, y) = F(\Phi(t), \Psi(t)) = F(\phi(t-s), \psi(t-s)) \] \[ G(x, y) = G(\Phi(t), \Psi(t)) = G(\phi(t-s), \psi(t-s)) \]

Check if the new functions satisfy the autonomous system

In this step, we will compare the derivatives we found in Step 1 with rewritten F and G functions from Step 2. We have: \[ \frac{dΦ}{dt} = \phi'(t-s) = F(\phi(t-s), \psi(t-s)) = F(\Phi(t), \Psi(t)) \] \[ \frac{dΨ}{dt} = \psi'(t-s) = G(\phi(t-s), \psi(t-s)) = G(\Phi(t), \Psi(t)) \] So, x = Φ(t) = φ(t - s) and y = Ψ(t) = ψ(t - s) satisfy the autonomous system.

Verify if the new solutions work for the given range of t

The original range of t is given by \(\alpha < t < \beta\). We need to show that the new solutions x = Φ(t) = φ(t - s) and y = Ψ(t) = ψ(t - s) satisfy the range \(\alpha + s < t < \beta + s\). Since φ(t) and ψ(t) are initially defined for the range \(\alpha < t < \beta\), we can rewrite the range for Φ(t) and Ψ(t) as follows: \[ \alpha < t - s < \beta \Rightarrow \alpha + s < t < \beta + s \] With this, we have shown that x = Φ(t) = φ(t - s) and y = Ψ(t) = ψ(t - s) is a solution for the given autonomous system for the range \(\alpha + s < t < \beta + s\) for any real number s.

Key concepts

Chain Rule

The chain rule is a vital concept in calculus used for differentiating composite functions. It helps us understand how changes in one variable affect another through a linked sequence of functions. In simpler terms, the chain rule allows us to take the derivative of a function that is defined in terms of another function.

Here's how it works: If you have a function that depends on another function—say, if \( y = f(g(x)) \)—the derivative of \( y \) with respect to \( x \) is found by multiplying the derivative of \( y \) with respect to \( g \) by the derivative of \( g \) with respect to \( x \). Mathematically, this is expressed as:
\[\frac{dy}{dx} = \frac{dy}{dg} \cdot \frac{dg}{dx}\]

In the context of the given problem, this rule is applied to find \( \frac{d\Phi}{dt} \) and \( \frac{d\Psi}{dt} \) by differentiating \( \phi(t-s) \) and \( \psi(t-s) \) with respect to \( t \).

• Since \( t-s \) is a straightforward subtraction, its derivative with respect to \( t \) is 1.
• The result is a simpler derivative that matches the functions \( F \) and \( G \).

Differential Equations

Differential equations are mathematical equations that relate some function with its derivatives. Essentially, they capture how a particular quantity changes over time or space.

In the problem at hand, we're dealing with an autonomous differential system:

• Autonomous systems have functions \( F(x, y) \) and \( G(x, y) \) that depend only on the variables \( x \) and \( y \), and not explicitly on time \( t \).
• This property implies that the system's behavior depends only on its current state and not on how it arrived there.

By working with the differential equations \( \frac{dx}{dt} = F(x, y) \) and \( \frac{dy}{dt} = G(x, y) \), we seek to find functions \( \phi(t) \) and \( \psi(t) \) that describe the evolution of \( x \) and \( y \) over time. The solution \( x = \phi(t-s) \) and \( y = \psi(t-s) \) demonstrates how these equations hold true even after a horizontal shift in time, ensuring they continue to satisfy the system.

Time Translation Invariance

Time translation invariance is a key feature of autonomous systems, where the equations governing the system's behavior do not change over time. This means the system behaves the same way, irrespective of the time reference we choose.

In practical terms, this invariance allows us to shift the solution of a differential equation system in time without altering its correctness. For instance:

• Original solutions are given by \( x = \phi(t) \) and \( y = \psi(t) \).
• By shifting the time reference to \( t-s \), we produce new solutions \( x = \Phi(t) = \phi(t-s) \) and \( y = \Psi(t) = \psi(t-s) \).
• Even with this shift, the structure of the equations remains unchanged as \( \frac{dx}{dt} = F(x, y) \) and \( \frac{dy}{dt} = G(x, y) \) continue to hold true.

This property simplifies the analysis of systems over different time intervals, ensuring consistency across shifts.