Question

In Problems 13-32 use variation of parameters to solve the given nonhomogeneous system.

$27.\mathrm{X}=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\mathrm{X}+\left(\begin{array}{c}0\\ sec\mathrm{t}tan\mathrm{t}\end{array}\right)$

Short answer

The general solution of$27.\mathrm{X}=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\mathrm{X}+\left(\begin{array}{c}0\\ sec\mathrm{t}tan\mathrm{t}\end{array}\right)$ is

$\mathrm{X}\left(\mathrm{t}\right)={\mathrm{c}}_1\left(\begin{array}{c}\cos \mathrm{t}\\ -\sin \mathrm{t}\end{array}\right)+{\mathrm{c}}_2\left(\begin{array}{c}\sin \mathrm{t}\\ \cos \mathrm{t}\end{array}\right)+\left(\begin{array}{c}\cos \mathrm{t}\\ -\sin \mathrm{t}\end{array}\right)\mathrm{t}+\left(\begin{array}{c}-\sin \mathrm{t}\\ \sin \mathrm{t}\tan \mathrm{t}\end{array}\right)-\left(\begin{array}{c}\sin \mathrm{t}\\ \cos \mathrm{t}\end{array}\right)\ln |\cos \mathrm{t}$

Step-by-step solution

Variation of parameters for nonhomogeneous linear systems:

Parameter variation is a generic approach for identifying a specific solution of a differential equation by substituting the constants in the solution of a related (homogeneous) equation with functions and defining these functions so that the original differential equation is fulfilled.

Find the characteristic equation of the coefficient matrix:

We have

$\mathrm{X}\text{'}=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\mathrm{X}+\left(\begin{array}{c}0\\ \sec \mathrm{t}\tan \mathrm{t}\end{array}\right)$

Where,

$\mathrm{A}=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$

Now, we find the characteristic equation of the coefficient matrix,

$$\begin{gathered} \det =(\mathrm{A}-\mathrm{\lambda l})=0\to \left(\begin{array}{cc}0-\mathrm{\lambda }& 1\\ -1& 0-\mathrm{\lambda }\end{array}\right)=0 \\ {\mathrm{\lambda }}^2+1=0 \\ {\mathrm{\lambda }}^2=-1 \\ \mathrm{\lambda }=\pm \mathrm{i} \end{gathered}$$

so our eigenvalues are ${\mathrm{\lambda }}_1=\mathrm{i}$, and ${\mathrm{\lambda }}_2=\overline{{\mathrm{\lambda }}_1}=-\mathrm{i}$.

Find the eigenvector and a corresponding solution vector:

For :

Apply row operation

so here we have a single equation,

choosing yields . This gives an eigenvector:

and column vectors:

, and

also,

Where and

Therefore,

and,

Hence, the complementary function is

Find the invertible matrix:

The entries in form the first column of , and the entries in form the second column of . Therefore,

we want to make sure that $\Phi(t)$ is an invertible matrix by checking the determinant, where

since the determinant does not equal zero, the matrix is in fact, an invertible matrix.

So now,

Find the general solution of a Nonhomogeneous system:

obtaining the particular solution,

Finally, the general solution of the system is