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

[Lambda96]

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

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?

 

[DaveE]

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.

 

[Lambda96]

Thanks DaveE for your help and the links