Question

Find all complex solutions of the system.

$\frac{\mathbf{d}\overrightarrow{\mathbf{x}}}{\mathbf{d}\mathbf{t}}\mathbf{=}\left[\begin{array}{cc}3& -2\\ 5& -3\end{array}\right]\overrightarrow{\mathbf{x}}$

In the form given in Theorem 9.2.3. What solution do you get if you let ${\mathit{c}}_{\mathbf{1}}\mathbf{=}{\mathit{c}}_{\mathbf{2}}\mathbf{=}\mathbf{1}$?

Short answer

Thus, the general solution is $\overrightarrow{x}\left(t\right)=c_1{e}^{-it}\left[\begin{array}{c}3+i\\ 5\end{array}\right]+c_2{e}^{-it}\left[\begin{array}{c}3-i\\ 5\end{array}\right]$and when $c_1=c_2=1$the solution of the system is .

Step-by-step solution

Determine explanation for the given system.

The continuous dynamical systems with complex eigenvalues theorem states.

Consider a linear system,

$\frac{d\overrightarrow{x}}{dt}=A\overrightarrow{x}$

Suppose there exists a complex Eigen basis $\overrightarrow{v_1},\dots ,\overrightarrow{v_n}$for A , with associated complex eigenvalues ${\lambda }_1,\dots ,{\lambda }_n$. Then the general complex solution of the system is:

$\overrightarrow{\mathbf{x}}\left(t\right)\mathbf{=}{\mathit{c}}_{\mathbf{1}}{\mathit{e}}^{{\mathbf{\lambda }}_{\mathbf{1}}\mathbf{t}}\overrightarrow{{\mathbf{v}}_{\mathbf{1}}}\mathbf{+}\mathbf{\dots }\mathbf{+}{\mathit{c}}_{\mathbf{n}}{\mathbf{e}}^{{\mathbf{\lambda }}_{\mathbf{n}}\mathbf{t}}\overrightarrow{{\mathbf{v}}_{\mathbf{n}}}$

Here, $c_i$are arbitrary complex numbers.

Calculate the Eigen values of given system.

Consider the system as follows:

$\frac{d\overrightarrow{x}}{dt}=\left[\begin{array}{cc}3& -2\\ 5& -3\end{array}\right]\overrightarrow{x}$

Determine the eigenvalues as follows:

$A=\left[\begin{array}{cc}3& -2\\ 5& -3\end{array}\right]$

Then, the characteristic equation of A is $det\left|A-\lambda I\right|=0$which gives;

$\begin{array}{rcl}det\left|\begin{array}{c}\left[\begin{array}{cc}3-\lambda & -2\\ 5& -3-\lambda \end{array}\right]\end{array}\right|& =& 0\\ -\left(3-\lambda \right)\left(3+\lambda \right)+10& =& 0\\ {\lambda }^2+1& =& 0\\ \lambda & =& \pm i\end{array}$

Obtain the solution of the given system.

The Eigen spaces that correspond to the Eigen values $\lambda =\pm i$are:

$\begin{array}{rcl}{E}_i& =& ker\left[\begin{array}{cc}3-i& -2\\ 5& -3-i\end{array}\right]\\ E_{-i}& =& span\left[\begin{array}{c}3+i\\ 5\end{array}\right]\\ E_{-i}& =& span\left[\begin{array}{c}3-i\\ 5\end{array}\right]\end{array}$

Hence, the general solution is $\begin{array}{rcl}\overrightarrow{x}\left(t\right)& =& c_1{e}^{it}\left[\begin{array}{c}3+i\\ 5\end{array}\right]+c_2{e}^{-it}\left[\begin{array}{c}3-i\\ 5\end{array}\right]\end{array}$.

Obtain the solution when c1=c2=1 of the given system.

Solve further to obtain general solution as follows:

$\begin{array}{rcl}\overrightarrow{x}\left(t\right)& =& e^{it}\left[\begin{array}{c}3+i\\ 5\end{array}\right]+e^{-it}\left[\begin{array}{c}3-i\\ 5\end{array}\right]\\ & =& \left(\cos t+i\sin t\right)\left[\begin{array}{c}3+i\\ 5\end{array}\right]+\left(\cos t-i\sin t\right)\left[\begin{array}{c}3-i\\ 5\end{array}\right]\\ & =& \left[\begin{array}{c}6\cos t-2\sin t\\ 10\cos t\end{array}\right]\end{array}$

Hence, when $\begin{array}{rcl}{c}_1& =& c_2=1\end{array}$the solution of the system is:

$\begin{array}{rcl}\overrightarrow{x}\left(t\right)& =& \left[\begin{array}{c}6\cos t-2\sin t\\ 10\cos t\end{array}\right]\end{array}$