Diff. EQ. system I.V.P. with complex cunjugates (Please check)

[VinnyCee]

Here is the problem:

$$X'\,=\,\left(\begin{array}{cc}-2 & 5 \\-2 & 4 \end{array}\right)\,X,\,\,\,\,X(0)\,=\,\left(\begin{array}{c} 1 \\ 0 \end{array}\right)$$

Here is what I have:

$$det(A\,-\,r\,I)\,=\,0$$

$$r^2\,-\,2r\,+\,2\,=\,0$$

$$r\,=\,-\frac{1}{2}\,\pm\,i,\,\,\,\,\lambda\,=\,-\frac{1}{2},\,\,\mu\,=\,1$$

This is where I get confused. I have two possibilities for the r-value, right? Namely, they are $r_1\,=\,-\frac{1}{2}\,+\,i$ and $r_2\,=\,-\frac{1}{2}\,-\,i$. Now I go back and use the $A\,-\,r_n\,I\,=\,0$ equation to figure a general solution for the system.

For $r_1$:

$$\left[A\,-\,(-\frac{1}{2}\,+\,i)\,I\right]\,\left(\begin{array}{c}\xi_1 \\\xi_2 \end{array}\right)\,=\,0$$

$$\left(\begin{array}{cc}-\frac{3}{2}\,-\,i & 5 \\-2 & \frac{9}{2}\,-\,i \end{array}\right)\,\left(\begin{array}{c}\xi_1 \\\xi_2 \end{array}\right)\,=\,0$$

$$\left(-\frac{3}{2}\,-\,i\right)\,\xi_1\,+\,5\,\xi_2\,=\,0$$

$$5\,\xi_2\,=\,\left(\frac{3}{2}\,+\,i\right)\,\xi_1$$

$$\xi_2\,=\,\left(\frac{3}{10}\,+\,\frac{1}{5}\,i\right)\,\xi_1$$

$$\xi^{(1)}\,=\,\left(\begin{array}{c}\ 1 \\\ \frac{3}{10}\,+\,\frac{1}{5}\,i \end{array}\right)$$

$$\xi^{(1)}\,=\,\left(\begin{array}{c}\ 1 \\\ \frac{3}{10} \end{array}\right)\,+\,\left(\begin{array}{c}\ 0 \\\ \frac{1}{5} \end{array}\right)\,i$$

Now, finally, here is the general solution for $r_1$:

$$X_1_c\,=\,C_1\,e^{-\frac{t}{2}}\,\left[\left(\begin{array}{c}\ 1 \\\ \frac{3}{10} \end{array}\right)\,cos\,t\,-\,\left(\begin{array}{c}\ 0 \\\ \frac{1}{5} \end{array}\right)\,sin\,t\right]\,+\,C_2\,e^{-\frac{t}{2}}\,\left[\left(\begin{array}{c}\ 1 \\\ \frac{3}{10} \end{array}\right)\,sin\,t\,+\,\left(\begin{array}{c}\ 0 \\\ \frac{1}{5} \end{array}\right)\,cos\,t\right]$$

Using the initial condition given to solve for c1 and c2, I get:

$$C_1\,=\,1,\,\,\,\,\,\,C_2\,=\,-\frac{3}{2}$$

Plugging back into the general EQ:

$$X_1\,=e^{-\frac{t}{2}}\,\left[\left(\begin{array}{c}\ 1 \\\ \frac{3}{10} \end{array}\right)\,cos\,t\,-\,\left(\begin{array}{c}\ 0 \\\ \frac{1}{5} \end{array}\right)\,sin\,t\right]\,+\,\left(-\frac{3}{2}\right)\,e^{-\frac{t}{2}}\,\left[\left(\begin{array}{c}\ 1 \\\ \frac{3}{10} \end{array}\right)\,sin\,t\,+\,\left(\begin{array}{c}\ 0 \\\ \frac{1}{5} \end{array}\right)\,cos\,t\right]$$

And simplifying:

$$X_1\,=\,e^{-\frac{t}{2}}\,\left(\begin{array}{c}\ cos\,t\,-\,\frac{3}{2}\,sin\,t \\\ -\frac{13}{20}\,sin\,t \end{array}\right)$$

Now here is where I am confused. I get this answer for $X_1$, and I get a similar one for $X_2$ using the $r_2$ complex conjugate and the initial values:

$$X_2\,=\,e^{-\frac{t}{2}}\,\left(\begin{array}{c}\ -2\,sin\,t \\\ -\frac{3}{5}\,sin\,t\,-\,\frac{2}{5}\,cos\,t \end{array}\right)$$

Which one is correct (if either)? Or are they somehow the same?

 

[ehild]

Here is the problem:

$$X'\,=\,\left(\begin{array}{cc}-2 & 5 \\-2 & 4 \end{array}\right)\,X,\,\,\,\,X(0)\,=\,\left(\begin{array}{c} 1 \\ 0 \end{array}\right)$$

Here is what I have:

$$det(A\,-\,r\,I)\,=\,0$$

$$r^2\,-\,2r\,+\,2\,=\,0$$

$$r\,=\,-\frac{1}{2}\,\pm\,i,\,\,\,\,\lambda\,=\,-\frac{1}{2},\,\,\mu\,=\,1$$

You've made a mistake.

$$r\,=\,1\,\pm\,i$$

ehild

 

[VinnyCee]

Thanks

Thanks for showing me, I would have never found it!

After correcting that, I find the following:

$$r_1\,=\,1\,+\,i$$

$$\left(\begin{array}{cc}-3\,-\,i & 5 \\-2 & 3\,-\,i \end{array}\right)\,\left(\begin{array}{c}\xi_1 \\\xi_2 \end{array}\right)\,=\,0$$

$$\xi^{(1)}\,=\,\left(\begin{array}{c}\ 1 \\\ \frac{3}{5} \end{array}\right)\,+\,\left(\begin{array}{c}\ 0 \\\ \frac{1}{5} \end{array}\right)\,i$$

$$X_1_c\,=\,C_1\,e^{t}\,\left[\left(\begin{array}{c}\ 1 \\\ \frac{3}{5} \end{array}\right)\,cos\,t\,-\,\left(\begin{array}{c}\ 0 \\\ \frac{1}{5} \end{array}\right)\,sin\,t\right]\,+\,C_2\,e^{t}\,\left[\left(\begin{array}{c}\ 1 \\\ \frac{3}{5} \end{array}\right)\,sin\,t\,+\,\left(\begin{array} {c}\ 0 \\\ \frac{1}{5} \end{array}\right)\,cos\,t\right]$$

$$C_1\,=\,1,\,\,\,\,\,\,C_2\,=\,-3$$

Assuming that is all correct, what do I do about the other three possibilities(two for each complex conjugate)?

 

[ehild]

Thanks for showing me, I would have never found it!

After correcting that, I find the following:

$$r_1\,=\,1\,+\,i$$

$$\left(\begin{array}{cc}-3\,-\,i & 5 \\-2 & 3\,-\,i \end{array}\right)\,\left(\begin{array}{c}\xi_1 \\\xi_2 \end{array}\right)\,=\,0$$

$$\xi^{(1)}\,=\,\left(\begin{array}{c}\ 1 \\\ \frac{3}{5} \end{array}\right)\,+\,\left(\begin{array}{c}\ 0 \\\ \frac{1}{5} \end{array}\right)\,i$$

The general solution of the system of differential equation is

$$X=c_1\xi(1)e^{r_1t}+c_2\xi(2)e^{r_2t}$$

You know the array $$\xi(1)$$ already, determine the other one belonging to the other root. Never mind that everything is complex :).
Now apply the initial condition and get c1 and c2. They will be complex (complex conjugates)
Now you can transform the exponentials to the trigonometric form and do all multiplications.
You get the solution and it is unique.

ehild

 

[VinnyCee]

?

Well, there are two roots that we found, $r_1$ and $r_2$.

Each of those has two equations from which to solve for $\xi$. I am really confused! Please help! I cannot just arbitrarily pick 2 out of 4 solutions and say it is right! What am I missing?

 

[VinnyCee]

Here are the four solutions...

$$\xi^{(1)}\,=\,\left(\begin{array}{c}\ 1 \\\ \frac{3}{5} \end{array}\right)\,+\,\left(\begin{array}{c}\ 0 \\\ \frac{1}{5} \end{array}\right)\,i$$

$$\xi^{(2)}\,=\,\left(\begin{array}{c}\ 1 \\\ \frac{3}{5} \end{array}\right)\,+\,\left(\begin{array}{c}\ 0 \\\ -\frac{1}{5} \end{array}\right)\,i$$

$$\xi^{(3)}\,=\,\left(\begin{array}{c}\ \frac{3}{2} \\\ 1 \end{array}\right)\,+\,\left(\begin{array}{c}\ \frac{1}{2} \\\ 0 \end{array}\right)\,i$$

$$\xi^{(4)}\,=\,\left(\begin{array}{c}\ \frac{3}{2} \\\ 1 \end{array}\right)\,+\,\left(\begin{array}{c}\ -\frac{1}{2} \\\ 0 \end{array}\right)\,i$$

But now what do I do with these?

 

[AKG]

This site is great. I found it very useful last semester for my ODE course.

 

[saltydog]

VinnyCee: When you have complex eigenvalues, you need compute the eigenvector corresponding to only one of the two complex eigenvalues. By breaking up the resulting complex-valued solution into its Real and Imaginary parts, you obtain a pair of independent solutions which together with two arbitrary constants, make up the general solution.

So for example, choose one eigenvalue, obtain one eigenvector and then obtain the complex-value solution:

$$Y(t)=Y_{re}(t)+iY_{im}(t)$$

Then the general solution is given by:

$$Y(t)=k_1Y_{re}(t)+k_2Y_{im}(t)$$

 

[VinnyCee]

Choosing one Eigenvalue

I choose the first one:

$$\xi^{(1)}\,=\,\left(\begin{array}{c}\ 1 \\\ \frac{3}{5} \end{array}\right)\,+\,\left(\begin{array}{c}\ 0 \\\ \frac{1}{5} \end{array}\right)\,i$$

But what do you mean by $Y(t)=Y_{re}(t)+iY_{im}(t)$?

Like this?:

$$Y(t)=\begin{array}{c}\ 1 \\\ \frac{3}{5} \end{array}\right)+...$$

 

[ehild]

Well, there are two roots that we found, $r_1$ and $r_2$.

Each of those has two equations from which to solve for $\xi$. I am really confused! Please help! I cannot just arbitrarily pick 2 out of 4 solutions and say it is right! What am I missing?

You have got two systems of equations, one for each root.
As they are systems of homogeneous equations, one of the components of the $\xi$ arrays is arbitrary. You can take it 1. Only the ratio of the components is defined by the equations.
So you have one array for r1 and an other one for r2.
For

$$r_1=1+i$$
$$\xi{(1)}\,=\,\left(\begin{array}{c}\ 1 \\\ \frac{3+i}{5} \end{array}\right)$$

For

$$r_2=1-i$$
$$\xi{(2)}\,=\,\left(\begin{array}{c}\ 1 \\\ \frac{3-i}{5} \end{array}\right)$$

The general solution is

$$X=c_1\xi(1)e^{(1+i)t} + c_2\xi(2)e^{(1-i)t}$$.

In components:

$$x_1=c_1e^{(1+i)t}+c_2e^{(1-i)t}=1$$

$$x_2= c_1\frac{3+i}{5}e^{(1+i)t}+c_2\frac{3-i}{5}e^{(1-i)t}$$

At t=0:

$$c_1+c_2 = 1$$

$$c_1\frac{3+i}{5}+c_2\frac{3-i}{5}=0$$

Solve for the c-s, replace them back to the general solution, use the trigonometric form, simplify, and you get (if I did everything correctly)

$$x_1= e^t(\cos{t} -3\sin{t})$$

and

$$x_2 = -2 e^t \sin{t}$$

ehild

 

[saltydog]

I choose the first one:

$$\xi^{(1)}\,=\,\left(\begin{array}{c}\ 1 \\\ \frac{3}{5} \end{array}\right)\,+\,\left(\begin{array}{c}\ 0 \\\ \frac{1}{5} \end{array}\right)\,i$$

But what do you mean by $Y(t)=Y_{re}(t)+iY_{im}(t)$?

Like this?:

$$Y(t)=\begin{array}{c}\ 1 \\\ \frac{3}{5} \end{array}\right)+...$$

VinnyCee, we have:

$$\lambda_1=1+i$$

The eigenvector corresponding to this eigenvalue is:

$$\left(<br /> \begin{array}{c}\ <br /> 1 \\<br /> \frac{1}{5}(3+i)\<br /> \end{array}<br /> \right)$$

Thus a solution is:

$$Y(t)=e^{(1+i)t}<br /> \left(<br /> \begin{array}{c}\ <br /> 1 \\<br /> \frac{1}{5}(3+i)\<br /> \end{array}<br /> \right)$$

Now, convert this to a real part and imaginary part via Euler's formula:

$$Y(t)=<br /> \left(<br /> \begin{array}{c}\ <br /> e^tCos(t) \\<br /> e^t[\frac{3}{5}Cos(t)-\frac{1}{5}Sin(t)]\<br /> \end{array}<br /> \right) +i<br /> <br /> \left(<br /> \begin{array}{c}\ <br /> e^tSin(t) \\<br /> e^t[\frac{3}{5}Sin(t)+\frac{1}{5}Cos(t)]\<br /> \end{array}<br /> \right)$$


Note how the real part and the imaginary part has been separated into separate terms which are usually written:

$$Y(t)=Y_{re}(t)+iY_{im}(t)$$

As I stated earlier, the general solution is then:

$$Y(t)=k_1Y_{re}(t)+k_2Y_{im}(t)$$

Substituting the initial conditions, I get the same answer as ehild which I checked via Mathematica by back substitution. A plot of both functions is attached.

 

[saltydog]

A qualitative analysis

The global behavior of linear systems like these is determined by the value of the eigenvalues.

For the case with complex eigenvalues, the real part determines the behavior of the phase portrait (when y is graphed as a function of x parametrically. If the real part is less than 0, the exponential term of the solution forces the portrait to spiral into the origin. This is a spirial sink.

If the real part is greater than zero, the solution spirials off away from the origin to infinity. This is a spirial source.

And if the real part is 0, the solution is periodic.

The attached plot exhibits this solution with complex eigenvalue that has real part equal to 1. The solution tends away from its source.

 

[VinnyCee]

Ok, but what about the E3 and E4 vectors that I also found?

That covers E1 and E2, but what about these?:

$$\xi^{(3)}\,=\,\left(\begin{array}{c}\ \frac{3}{2} \\\ 1 \end{array}\right)\,+\,\left(\begin{array}{c}\ \frac{1}{2} \\\ 0 \end{array}\right)\,i$$

$$\xi^{(4)}\,=\,\left(\begin{array}{c}\ \frac{3}{2} \\\ 1 \end{array}\right)\,+\,\left(\begin{array}{c}\ -\frac{1}{2} \\\ 0 \end{array}\right)\,i$$

 

[saltydog]

That covers E1 and E2, but what about these?:

$$\xi^{(3)}\,=\,\left(\begin{array}{c}\ \frac{3}{2} \\\ 1 \end{array}\right)\,+\,\left(\begin{array}{c}\ \frac{1}{2} \\\ 0 \end{array}\right)\,i$$

$$\xi^{(4)}\,=\,\left(\begin{array}{c}\ \frac{3}{2} \\\ 1 \end{array}\right)\,+\,\left(\begin{array}{c}\ -\frac{1}{2} \\\ 0 \end{array}\right)\,i$$

Dude, I'll be honest with you: I don't know where you gettin' those [itex]\xi's[/tex] but I bet a dollar the answer I gave you is the general solution,$\xi's$ or no $\xi's$. <img src="https://cdn.jsdelivr.net/joypixels/assets/8.0/png/unicode/64/1f642.png" class="smilie smilie--emoji" loading="lazy" width="64" height="64" alt=":smile:" title="Smile :smile:" data-smilie="1"data-shortname=":smile:" />[/itex]

 

[VinnyCee]

Sweet! Almost done.

So the general solution is:

$$X(t)\,=\,C_1\,e^t\,\left[\left(\begin{array}{c}\ 1 \\\ \frac{3}{5} \end{array}\right)\,cos\,t\,-\,\left(\begin{array}{c}\ 0 \\\ \frac{1}{5} \end{array}\right)\,sin\,t\right]\,+\,C_2\,e^t\,\left[\left(\begin{array}{c}\ 1 \\\ \frac{3}{5} \end{array}\right)\,sin\,t\,+\,\left(\begin{array}{c}\ 0 \\\ \frac{1}{5} \end{array}\right)\,cos\,t\right]$$

And then solving for the initial condition, I get $C_1\,=\,1$ and $C_2\,=\,-3$, right?

So then the exact solution is:

$$X(t)\,=\,e^t\,\left[\left(\begin{array}{c}\ 1 \\\ \frac{3}{5} \end{array}\right)\,cos\,t\,-\,\left(\begin{array}{c}\ 0 \\\ \frac{1}{5} \end{array}\right)\,sin\,t\right]\,-\,3\,e^t\,\left[\left(\begin{array}{c}\ 1 \\\ \frac{3}{5} \end{array}\right)\,sin\,t\,+\,\left(\begin{array}{c}\ 0 \\\ \frac{1}{5} \end{array}\right)\,cos\,t\right]$$

Is that right?

 

[VinnyCee]

Now I have simplified(hopefully!) that answer down to:

$$X(t)\,=\,e^t\,\left(\begin{array}{c}\ cos\,t\,-\,3\,sin\,t \\\ -2\,sin\,t \end{array}\right)$$

Is this all correct now? Thanks a lot for the help saltydog:)

 

[saltydog]

Now I have simplified(hopefully!) that answer down to:

$$X(t)\,=\,e^t\,\left(\begin{array}{c}\ cos\,t\,-\,3\,sin\,t \\\ -2\,sin\,t \end{array}\right)$$

Is this all correct now? Thanks a lot for the help saltydog:)

Yes. Ehild got it before me.