6.5 Solutions to Homogeneous Systems

A. Fundamental Solution Set and Wronskian

We start with studying the homogeneous linear system

 [asciimath]bb(y)'=Abb(y)[/asciimath](6.5.1)

where [asciimath]A[/asciimath] is an [asciimath]nxxn[/asciimath] constant matrix with real entries. [asciimath]bb"y" = bb"0"[/asciimath] is the trivial solution of the system. Any other solution is a nontrivial solution.

Theorem. If [asciimath]bb"y"_1, bb"y"_2, ..., bb"y"_n[/asciimath] are [asciimath]n[/asciimath] linearly independent solutions to the system 6.5.1 and [asciimath]A[/asciimath] is continuous on an open interval [asciimath]I[/asciimath], then the set [asciimath]{bb"y"_1, bb"y"_2, ..., bb"y"_n}[/asciimath] is called a fundamental set of solutions to the system on [asciimath]I[/asciimath].

Revisiting Section 6.2 on linear independence, vectors [asciimath]bb"y"_1, bb"y"_2, ..., bb"y"_n[/asciimath] are linearly independent if  [asciimath]c_1bb"y"_1+c_2bb"y"_2+...+c_nbb"y"_n=0[/asciimath]  has only the trivial solution. That is if

  [asciimath]c_1bb"y"_1+c_2bb"y"_2+...+c_nbb"y"_n=[/asciimath]  [asciimath][[y_11, y_12 ,..., y_(1n)],[y_21,y_22, ..., y_(2n)], [vdots,vdots ,ddots, vdots] ,[y_(n1), y_(n2) , ..., y_(n\n)] ][/asciimath]  [asciimath][[c_1],[c_2], [vdots] ,[c_n] ]=bb"0"[/asciimath]

then  [asciimath]bb"c"=[[c_1],[c_2], [vdots] ,[c_n] ]=bb"0"[/asciimath]

For this to be the only solution (unique solution), the determinant of the matrix of coefficient of the equation whose columns are the vector functions  [asciimath][(bb"y"_1,bb"y"_2, ..., bb"y"_n)][/asciimath] must be nonzero. The determinant of the matrix of coefficient of the equation is called the Wronskian and denoted [asciimath]W(t)[/asciimath].

 [asciimath]W(t)=|bb"y"_1 \ bb"y"_2 \ ... \ bb"y"_n|[/asciimath]

Theorem. If the Wronkian [asciimath]W(t)[/asciimath] of [asciimath]{bb"y"_1, bb"y"_2, ..., bb"y"_n}[/asciimath] is nonzero at some point (and thus never zero) on [asciimath]I[/asciimath], then [asciimath]{bb"y"_1, bb"y"_2, ..., bb"y"_n}[/asciimath] is linearly independent, forming a fundamental solution set for system 6.5.1 on [asciimath]I[/asciimath]. The fundamental matrix [asciimath]Y(t)[/asciimath] of the system is

 [asciimath]bb"Y"(t)=[/asciimath] [asciimath][(bb"y"_1,bb"y"_2, ..., bb"y"_n)]=[/asciimath] [asciimath][[y_11, y_12 ,..., y_(1n)],[y_21,y_22, ..., y_(2n)], [vdots,vdots ,ddots, vdots] ,[y_(n1), y_(n2) , ..., y_(n\n)] ][/asciimath]

B. Solutions to Homogeneous Systems with Constant Coefficients

In our quest to find solutions to homogeneous systems with constant coefficients, represented by system 6.5.1, we apply a similar approach to that used in solving homogeneous linear differential equations with constant coefficients.

Recall from Section 3.2 that we guessed a nontrivial solution of the form  [asciimath]y = e^(rt)[/asciimath] for a homogeneous linear differential equation with constant coefficients. Section 6.4 showed that any higher-order linear differential equation can be expressed as a first-order linear system of differential equations. Therefore, it is reasonable that a solution for system 6.5.1 to be of the form

 [asciimath]mathbf(y) = e^(rt)mathbf(u)[/asciimath](6.5.2)

Here, [asciimath]r[/asciimath] is a constant, and [asciimath]mathbf(u)[/asciimath] is a constant vector. The next step is to substitute the guessed solution 6.5.2 into our system. Doing so gives

 [asciimath]bb"y"'=Abb"y"[/asciimath]

 [asciimath]re^(rt)mathbf(u) = Ae^(rt)mathbf(u)[/asciimath]

After canceling the exponential term  [asciimath]e^(rt)[/asciimath], we arrive at

 [asciimath]rmathbf(u) = Amathbf(u)[/asciimath]

Rearranging this equation leads to

 [asciimath](A - rI)mathbf(u) = 0[/asciimath]

This is the characteristic equation used to find the eigenvalues and eigenvectors of matrix [asciimath]A[/asciimath], as seen in Section 6.3. For our guessed solution [asciimath]mathbf(y) = e^(rt)mathbf(u)[/asciimath] to be nontrivial, [asciimath]r[/asciimath] and [asciimath]mathbf(u)[/asciimath]  must correspond to the eigenvalue and eigenvector of matrix [asciimath]A[/asciimath], respectively.

Therefore, to solve system 6.5.1, we first find the eigenvalues and eigenvectors of the coefficient matrix [asciimath]A[/asciimath]. The solution structure varies depending on the nature of the eigenvalues, which can be real and distinct, complex, or repeated. Each of these scenarios will be explored in the following sections.

Section 6.5 Exercises