Question

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

Short answer

The general solution of is

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:

Where

And

Obtaining the eigenvalues of the coefficient matrix,

So our eigenvalues are and

Find the eigenvector and a corresponding solution vector:

From the third row, we see that ,and from the first row we have

Choosing yields . This gives an eigenvector and a corresponding solution vector

:

From the third row, we see that , , and from the second row we have

Choosing yields . This gives an eigenvector and a corresponding solution vector

Apply row operation

From the second row, we see that , and from the first row we have

As forwe will choose an arbitrary value, say . This gives an eigenvector and a corresponding solution vector

Hence, the complementary solution is

Find the invertible matrix:

The entries in form the first column of and the entries in form the second column of and the entries in form the third 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 we find the inverse of the matrix , since this is a matrix, finding the inverse will be a bit lengthy, first we transpose the original matrix

Then we find the determinant of each of the minor matrices.

Find the general solution of a Nonhomogeneous system:

Now we create the matrix of cofactors, we must then reverse the sign of alternating terms of this new matrix,

Finally,

Obtaining the particular solution,

Therefore, the general solution of the system is