Solve these two coupled first-order differential equations and sketch the flow

In summary, DaveE was helping another user solve a 2nd order differential equation. The user had substituted x(t) into the equation to obtain a 2nd order differential equation and then used the Ansatz y(t)=e^{\lambda t} to obtain the solution. They then used the solution to obtain x(t). Unfortunately, they were not sure about task b and were wondering if they needed to plot only the real part or the complex part of the solution. They looked for features like where the derivative(s) were zero to find stable points and asymptotes.
  • #1
Lambda96
186
64
Homework Statement
solve the two coupled first-order differential equations ##\textbf{f}(\textbf{x}(t))## and sketch the flow ##\phi_t(\textbf{x})##
Relevant Equations
none
Hi,

unfortunately, I have a problem to solve the following task

Bildschirmfoto 2023-07-04 um 20.08.10.png


The equation looks like this:

$$\left(\begin{array}{c} \frac{d}{dt} x(t) \\ \frac{d}{dt} y(t) \end{array}\right)=\left(\begin{array}{c} -a y(t) \\ x(t) \end{array}\right)$$

Since the following is true ##\frac{d}{dt} y(t)=x(t)## I substituted ##x(t)## into the first equation on the left hand side, obtaining a 2nd order differential equation, i.e. ##y''(t)=-ay(t)## to solve this differential equation I then used the Ansatz ##y(t)=e^{\lambda t}## and obtained the following solution.

$$y(t)=c_1 \ e^{i \sqrt{a}t}-c_2 \ e^{-i \sqrt{a}t}$$

I then obtain x(t) using ##\frac{d}{dt} y(t)=x(t)##

$$y(t)=i \ \sqrt{a} \ c_1 \ e^{i \sqrt{a}t}-i \ \sqrt{a} \ c_2 \ e^{-i \sqrt{a}t}$$

Unfortunately, I am now a bit unsure about task b. For the flow ##\phi_t(\textbf{x})## I would now simply draw the vector ##\textbf{f}(\textbf{x}(t))=\left(\begin{array}{c} -a y(t) \\ x(t) \end{array}\right)## with my solution from task a for ##y(t)## and ##x(t)##.

Unfortunately I have problems to draw the vector, because I don't know what the constants ##c_1## and ##c_2## are and unfortunately my solution from task a is complex, so do I have to plot only the real part or is my solution from task a wrong?
 
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  • #2
The flow map doesn't assume one particular set of ICs. That would make only one line in the flow map.

Since ##a \in \mathbb R ## your solutions should be real too. This is a physically realizable system. This is a constraint on ##c_1## and ##c_2##. Your ICs will be various values of ##x(0)## and ##y(0)## which will then determine ##c_1## and ##c_2##.

This may help:
http://www.math.sjsu.edu/~simic/Fall05/Math134/flows.pdf

But there's lots of stuff on the web. Steve Strogatz has a nice set of lectures on YouTube in dynamic systems that does flow maps, but it's long.

Look for features like where the derivative(s) are zero to find stable points and asymptotes.

Sorry, it's been a long time since I really did this sort of work, and I'm too lazy to actually solve this system.
 
Last edited:
  • Informative
Likes berkeman
  • #3
Thanks DaveE for your help and the links 👍👍
 
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Likes berkeman and DaveE

Related to Solve these two coupled first-order differential equations and sketch the flow

What are coupled first-order differential equations?

Coupled first-order differential equations are a set of two or more differential equations that involve multiple dependent variables and their derivatives, where the equations are linked or "coupled" together through these variables. Solving them requires finding functions for these variables that satisfy all the equations simultaneously.

How do you solve coupled first-order differential equations?

To solve coupled first-order differential equations, you can use methods such as substitution, matrix algebra, or numerical techniques. Analytical solutions might involve expressing one variable in terms of another and then solving the resulting equations. For numerical solutions, methods like Euler's method or the Runge-Kutta method are commonly used.

What is the importance of initial conditions in solving these equations?

Initial conditions are crucial because they provide the specific values of the dependent variables at a given point, which are necessary to find a unique solution to the differential equations. Without initial conditions, you might end up with a general solution that includes arbitrary constants.

How do you sketch the flow of coupled differential equations?

To sketch the flow of coupled differential equations, you typically plot the trajectories of the solutions in a phase plane, which is a graphical representation of the variables against each other. This involves solving the equations and plotting the resulting curves, which show how the variables evolve over time. Tools like direction fields or vector fields can also be helpful in visualizing the flow.

Can software tools be used to solve and visualize coupled differential equations?

Yes, software tools like MATLAB, Mathematica, and Python (with libraries such as NumPy and SciPy) can be used to solve and visualize coupled differential equations. These tools offer both numerical solvers and visualization capabilities, making it easier to handle complex systems and generate phase plane plots.

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