Linear Homogeneous Systems of Differential Equations with Constant Coefficients


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Linear Homogeneous Systems of Differential Equations with Constant Coefficients

An \(n\)th order linear system of differential equations with constant coefficients is written as

\[\frac{{d{x_i}}}{{dt}} = {x'_i} = \sum\limits_{j = 1}^n {{a_{ij}}{x_j}\left( t \right)} + {f_i}\left( t \right),\;\; i = 1,2, \ldots ,n,\]

where \({x_1}\left( t \right),{x_2}\left( t \right), \ldots ,{x_n}\left( t \right)\) are unknown functions of the variable \(t,\) which often has the meaning of time, \({a_{ij}}\) are certain constant coefficients, which can be either real or complex, \({f_i}\left( t \right)\) are given (in general case, complex-valued) functions of the variable \(t.\)

We assume that all these functions are continuous on an interval \(\left[ {a,b} \right]\) of the real number axis \(t.\)

By setting

\[X\left( t \right) = \left[ {\begin{array}{*{20}{c}} {{x_1}\left( t \right)}\\ {{x_2}\left( t \right)}\\ \vdots \\ {{x_n}\left( t \right)} \end{array}} \right],\;\; X'\left( t \right) = \left[ {\begin{array}{*{20}{c}} {{x'_1}\left( t \right)}\\ {{x'_2}\left( t \right)}\\ \vdots \\ {{x'_n}\left( t \right)} \end{array}} \right],\;\; f\left( t \right) = \left[ {\begin{array}{*{20}{c}} {{f_1}\left( t \right)}\\ {{f_2}\left( t \right)}\\ \vdots \\ {{f_n}\left( t \right)} \end{array}} \right],\;\; A = \left[ {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}& \cdots &{{a_{1n}}}\\ {{a_{21}}}&{{a_{22}}}& \cdots &{{a_{2n}}}\\ \cdots & \cdots & \cdots & \cdots \\ {{a_{n1}}}&{{a_{n2}}}& \cdots &{{a_{nn}}} \end{array}} \right],\]

the system of differential equations can be written in matrix form:

\[X'\left( t \right) = AX\left( t \right) + f\left( t \right).\]

If the vector \(f\left( t \right)\) is identically equal to zero: \(f\left( t \right) \equiv 0,\) then the system is said to be homogeneous:

\[X'\left( t \right) = AX\left( t \right).\]

Homogeneous systems of equations with constant coefficients can be solved in different ways. The following methods are the most commonly used:

Below on this page we will discuss in detail the elimination method. Other methods for solving systems of equations are considered separately in the following pages.

Elimination Method

Using the method of elimination, a normal linear system of \(n\) equations can be reduced to a single linear equation of \(n\)th order. This method is useful for simple systems, especially for systems of order \(2.\)

Consider a homogeneous system of two equations with constant coefficients:

\[\left\{ \begin{array}{l} {x'_1} = {a_{11}}{x_1} + {a_{12}}{x_2}\\ {x'_2} = {a_{21}}{x_1} + {a_{22}}{x_2} \end{array} \right.,\]

where the functions \({x_1},{x_2}\) depend on the variable \(t.\)

We differentiate the first equation and substitute the derivative \({x'_2}\) from the second equation:

\[{x^{\prime\prime}_1} = {a_{11}}{x'_1} + {a_{12}}{x'_2},\;\; \Rightarrow {x^{\prime\prime}_1} = {a_{11}}{x'_1} + {a_{12}}\left( {{a_{21}}{x_1} + {a_{22}}{x_2}} \right),\;\; \Rightarrow {x^{\prime\prime}_1} = {a_{11}}{x'_1} + {a_{12}}{a_{21}}{x_1} + {a_{22}}{a_{12}}{x_2}.\]

Now we substitute \({a_{12}}{x_2}\) from the first equation. As a result we obtain a second order linear homogeneous equation:

\[{x^{\prime\prime}_1} = {a_{11}}{x'_1} + {a_{12}}{a_{21}}{x_1} + {a_{22}}\left( {{x'_1} - {a_{11}}{x_1}} \right),\;\; \Rightarrow {x^{\prime\prime}_1} = {a_{11}}{x'_1} + {a_{12}}{a_{21}}{x_1} + {a_{22}}{x'_1} - {a_{11}}{a_{22}}{x_1},\;\; \Rightarrow {x^{\prime\prime}_1} - \left( {{a_{11}} + {a_{22}}} \right){x'_1} + \left( {{a_{11}}{a_{22}} - {a_{12}}{a_{21}}} \right){x_1} = 0.\]

It is easy to construct its solution, if we know the roots of the characteristic equation:

\[{\lambda ^2} - \left( {{a_{11}} + {a_{22}}} \right)\lambda + \left( {{a_{11}}{a_{22}} - {a_{12}}{a_{21}}} \right) = 0.\]

In the case of real coefficients \({a_{ij}},\) the roots can be both real (distinct or multiple) and complex. In particular, if the coefficients \({a_{12}}\) and \({a_{21}}\) have the same sign, then the discriminant of the characteristic equation will always be positive and, therefore, the roots will be real and distinct.

After the function \({x_1}\left( t \right)\) is determined, the other function \({x_2}\left( t \right)\) can be found from the first equation.

The elimination method can be applied not only to homogeneous linear systems. It can also be used for solving nonhomogeneous systems of differential equations or systems of equations with variable coefficients.