Question

Solve the given system of differential equations by systematic elimination.

$$\begin{gathered} \mathit{D}\mathit{x}\mathbf{=}\mathit{y} \\ \mathit{D}\mathit{y}\mathbf{=}\mathit{z} \\ \mathit{D}\mathit{z}\mathbf{=}\mathit{x} \end{gathered}$$

Short answer

$$\begin{gathered} x\left(t\right)=c_1{e}^t+c_2{e}^{-\frac{1}{2}t}cos\frac{\sqrt{3}}{2}t+c_3{e}^{-\frac{1}{2}t}sin\frac{\sqrt{3}}{2}t \\ y\left(t\right)=c_1{e}^t+\left(-\frac{1}{2}{c}_2+\frac{\sqrt{3}}{2}{c}_3\right)e^{-\frac{1}{2}t}cos\frac{\sqrt{3}}{2}t+\left(-\frac{\sqrt{3}}{2}{c}_2-\frac{1}{2}{c}_3\right)e^{-\frac{1}{2}t}sin\frac{\sqrt{3}}{2}t \\ z\left(t\right)=c_1{e}^t+\left(-\frac{1}{2}{c}_2-\frac{\sqrt{3}}{2}{c}_3\right)e^{-\frac{1}{2}t}cos\frac{\sqrt{3}}{2}t+\left(\frac{\sqrt{3}}{2}{c}_2-\frac{1}{2}{c}_3\right)e^{-\frac{1}{2}t}sin\frac{\sqrt{3}}{2}t \end{gathered}$$

Step-by-step solution

Solution for X

We have the system of differential equations

$$\begin{gathered} Dx=y \\ Dy=z \\ Dz=x \end{gathered}$$

and we have to solve it using systematic elimination as the following technique:

Operate the first equation by the operator D and multiply the second equation by 1, then we obtain

$$\begin{gathered} D^{2}x-Dy=0 \\ Dy=z \end{gathered}$$

After that, add equation (4) to equation (5) and simplify, then we obtain

$$\begin{gathered} \left(D^{2}x-Dy\right)+\left(Dy\right)=z \\ (Dy-Dy)+\left(D^{2}x\right)=z \\ {D}^{2}x=z \end{gathered}$$

After that, operate equation (6) by the operator D, then we obtain

$D^{3}x-Dz=0$

Then add equation (7) to equation (3) and simplify, then we obtain

$$\begin{gathered} \left(D^{3}x-Dz\right)+\left(Dz\right)=x \\ (Dz-Dz)+\left(D^{3}x\right)=x \\ {D}^{3}x-x=0 \\ \left(D^3-1\right)x=0 \end{gathered}$$

After assuming that $x=e^{mt}$as a solution for our system, we can have the auxiliary solution for x as

$m^3-1=0$

$(m-1)\left(m^2+m+1\right)=0$

which has the roots

$m_1=1and{m}_{2,3}=\frac{-1\pm \sqrt{1-4\times 1\times 1}}{2}=-\frac{1}{2}\pm \frac{\sqrt{3}}{2}i$

Then we obtain

$x\left(t\right)=c_1{e}^t+c_2{e}^{-\frac{1}{2}t}cos\frac{\sqrt{3}}{2}t+c_3{e}^{-\frac{1}{2}t}sin\frac{\sqrt{3}}{2}t$

is the solution for x

Solutions for y

Now, we can obtain the solution for y by substituting with the solution for x into equation (1), then we obtain

$y\left(t\right)=Dx=D\left(c_1{e}^t+c_2{e}^{-\frac{1}{2}t}cos\frac{\sqrt{3}}{2}t+c_3{e}^{-\frac{1}{2}t}sin\frac{\sqrt{3}}{2}t\right)$

Then

$$\begin{gathered} y\left(t\right)=c_1{e}^t-\frac{1}{2}{c}_2{e}^{-\frac{1}{2}t}cos\frac{\sqrt{3}}{2}t-\frac{\sqrt{3}}{2}{c}_2{e}^{-\frac{1}{2}t}sin\frac{\sqrt{3}}{2}t-\frac{1}{2}{c}_3{e}^{-\frac{1}{2}t}sin\frac{\sqrt{3}}{2}t+\backslash \mathrm{hfill} \\ \frac{\sqrt{3}}{2}{c}_3{e}^{-\frac{1}{2}t}cos\frac{\sqrt{3}}{2}t \end{gathered}$$

Then,

$y\left(t\right)=c_1{e}^t+\left(-\frac{1}{2}{c}_2+\frac{\sqrt{3}}{2}{c}_3\right)e^{-\frac{1}{2}t}cos\frac{\sqrt{3}}{2}t+\left(-\frac{\sqrt{3}}{2}{c}_2-\frac{1}{2}{c}_3\right)e^{-\frac{1}{2}t}sin\frac{\sqrt{3}}{2}t$

Solution for z

Now, we can obtain the solution for z by substituting with the solution for y into equation (2), then we obtain

$z\left(t\right)=Dy=D\left(c_1{e}^t+\left(-\frac{1}{2}{c}_2+\frac{\sqrt{3}}{2}{c}_3\right)e^{-\frac{1}{2}t}cos\frac{\sqrt{3}}{2}t+\left(-\frac{\sqrt{3}}{2}{c}_2-\frac{1}{2}{c}_3\right)e^{-\frac{1}{2}t}sin\frac{\sqrt{3}}{2}t\right)$

Then

$$\begin{gathered} z\left(t\right)=c_1{e}^t-\frac{1}{2}\left(-\frac{1}{2}{c}_2+\frac{\sqrt{3}}{2}{c}_3\right)e^{-\frac{1}{2}t}cos\frac{\sqrt{3}}{2}t-\frac{\sqrt{3}}{2}\left(-\frac{1}{2}{c}_2+\frac{\sqrt{3}}{2}{c}_3\right)e^{-\frac{1}{2}t}sin\frac{\sqrt{3}}{2}t- \\ \frac{1}{2}\left(-\frac{\sqrt{3}}{2}{c}_2-\frac{1}{2}{c}_3\right)e^{-\frac{1}{2}t}sin\frac{\sqrt{3}}{2}t+\frac{\sqrt{3}}{2}\left(-\frac{\sqrt{3}}{2}{c}_2-\frac{1}{2}{c}_3\right)e^{-\frac{1}{2}t}cos\frac{\sqrt{3}}{2}t \end{gathered}$$

Then,

$z\left(t\right)=c_1{e}^t+\left(-\frac{1}{2}{c}_2-\frac{\sqrt{3}}{2}{c}_3\right)e^{-\frac{1}{2}t}cos\frac{\sqrt{3}}{2}t+\left(\frac{\sqrt{3}}{2}{c}_2-\frac{1}{2}{c}_3\right)e^{-\frac{1}{2}t}sin\frac{\sqrt{3}}{2}t$

Final Answers

$$\begin{gathered} x\left(t\right)=c_1{e}^t+c_2{e}^{-\frac{1}{2}t}cos\frac{\sqrt{3}}{2}t+c_3{e}^{-\frac{1}{2}t}sin\frac{\sqrt{3}}{2}t \\ y\left(t\right)=c_1{e}^t+\left(-\frac{1}{2}{c}_2+\frac{\sqrt{3}}{2}{c}_3\right)e^{-\frac{1}{2}t}cos\frac{\sqrt{3}}{2}t+\left(-\frac{\sqrt{3}}{2}{c}_2-\frac{1}{2}{c}_3\right)e^{-\frac{1}{2}t}sin\frac{\sqrt{3}}{2}t \\ z\left(t\right)=c_1{e}^t+\left(-\frac{1}{2}{c}_2-\frac{\sqrt{3}}{2}{c}_3\right)e^{-\frac{1}{2}t}cos\frac{\sqrt{3}}{2}t+\left(\frac{\sqrt{3}}{2}{c}_2-\frac{1}{2}{c}_3\right)e^{-\frac{1}{2}t}sin\frac{\sqrt{3}}{2}t \end{gathered}$$