Linear Nonhomogeneous Systems of Differential Equations with Constant Coefficients

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A normal linear inhomogeneous system of \(n\) equations with constant coefficients can be 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 \(t\) is the independent variable (often \(t\) is time), \({{x_i}\left( t \right)}\) are unknown functions which are continuous and differentiable on an interval \(\left[ {a,b} \right]\) of the real number axis \(t,\) \({a_{ij}}\left( {i,j = 1, \ldots ,n} \right)\) are the constant coefficients, \({f_i}\left( t \right)\) are given functions of the independent variable \(t.\) We assume that the functions \({{x_i}\left( t \right)},\) \({{f_i}\left( t \right)}\) and the coefficients \({a_{ij}}\) may take both real and complex values.

We introduce the following vectors:

\[\mathbf{X}\left( t \right) = \left[ {\begin{array}{*{20}{c}} {{x_1}\left( t \right)}\\ {{x_2}\left( t \right)}\\ \vdots \\ {{x_n}\left( t \right)} \end{array}} \right],\;\; \mathbf{f}\left( t \right) = \left[ {\begin{array}{*{20}{c}} {{f_1}\left( t \right)}\\ {{f_2}\left( t \right)}\\ \vdots \\ {{f_n}\left( t \right)} \end{array}} \right]\]

and the square matrix

\[A = \left[ {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}& \vdots &{{a_{1n}}}\\ {{a_{21}}}&{{a_{22}}}& \vdots &{{a_{2n}}}\\ \cdots & \cdots & \cdots & \cdots \\ {{a_{n1}}}&{{a_{n2}}}& \vdots &{{a_{nn}}} \end{array}} \right].\]

Then the system of equations can be written in a more compact matrix form as

\[\mathbf{X}'\left( t \right) = A\mathbf{X}\left( t \right) + \mathbf{f}\left( t \right).\]

For nonhomogeneous linear systems, as well as in the case of a linear homogeneous equation, the following important theorem is valid:

The general solution \(\mathbf{X}\left( t \right)\) of the nonhomogeneous system is the sum of the general solution \({\mathbf{X}_0}\left( t \right)\) of the associated homogeneous system and a particular solution \({\mathbf{X}_1}\left( t \right)\) of the nonhomogeneous system:

\[\mathbf{X}\left( t \right) = {\mathbf{X}_0}\left( t \right) + {\mathbf{X}_1}\left( t \right).\]

Methods of solutions of the homogeneous systems are considered on other web-pages of this section. Therefore, below we focus primarily on how to find a particular solution.

Another important property of linear inhomogeneous systems is the principle of superposition, which is formulated as follows:

If \({\mathbf{X}_1}\left( t \right)\) is a solution of the system with the inhomogeneous part \({\mathbf{f}_1}\left( t \right),\) and \({\mathbf{X}_2}\left( t \right)\) is a solution of the same system with the inhomogeneous part \({\mathbf{f}_2}\left( t \right),\) then the vector function

\[\mathbf{X}\left( t \right) = {\mathbf{X}_1}\left( t \right) + {\mathbf{X}_2}\left( t \right)\]

is a solution of the system with the inhomogeneous part

\[\mathbf{f}\left( t \right) = {\mathbf{f}_1}\left( t \right) + {\mathbf{f}_2}\left( t \right).\]

The most common methods of solution of the nonhomogeneous systems are the method of elimination, the method of undetermined coefficients (in the case where the function \(\mathbf{f}\left( t \right)\) is a vector quasi-polynomial), and the method of variation of parameters. Consider these methods in more detail.

Elimination Method

This method allows to reduce the normal nonhomogeneous system of \(n\) equations to a single equation of \(n\)th order. This method is useful for solving systems of order \(2.\)

Method of Undetermined Coefficients

The method of undetermined coefficients is well suited for solving systems of equations, the inhomogeneous part of which is a quasi-polynomial.

A real vector quasi-polynomial is a vector function of the form

\[\mathbf{f}\left( t \right) = {e^{\alpha t}}\left[ {\cos \left( {\beta t} \right){\mathbf{P}_m}\left( t \right) + \sin \left( {\beta t} \right){\mathbf{Q}_m}\left( t \right)} \right],\]

where \(\alpha,\) \(\beta\) are given real numbers, and \({{\mathbf{P}_m}\left( t \right)},\) \({{\mathbf{Q}_m}\left( t \right)}\) are vector polynomials of degree \(m.\) For example, a vector polynomial \({{\mathbf{P}_m}\left( t \right)}\) is written as

\[{\mathbf{P}_m}\left( t \right) = {\mathbf{A}_0} + {\mathbf{A}_1}t + {\mathbf{A}_2}{t^2} + \cdots + {\mathbf{A}_m}{t^m},\]

where \({\mathbf{A}_0},\) \({\mathbf{A}_2}, \ldots ,\) \({\mathbf{A}_m}\) are \(n\)-dimensional vectors (\(n\) is the number of equations in the system).

In the case when the inhomogeneous part \(\mathbf{f}\left( t \right)\) is a vector quasi-polynomial, a particular solution is also given by a vector quasi-polynomial, similar in structure to \(\mathbf{f}\left( t \right).\)

For example, if the nonhomogeneous function is

\[\mathbf{f}\left( t \right) = {e^{\alpha t}}{\mathbf{P}_m}\left( t \right),\]

a particular solution should be sought in the form

\[{\mathbf{X}_1}\left( t \right) = {e^{\alpha t}}{\mathbf{P}_{m + k}}\left( t \right),\]

where \(k = 0\) in the non-resonance case, i.e. when the index \(\alpha\) in the exponential function does not coincide with an eigenvalue \({\lambda _i}.\) If the index \(\alpha\) coincides with an eigenvalue \({\lambda _i},\) i.e. in the so-called resonance case, the value of \(k\) is chosen to be equal to the greatest length of the Jordan chain for the eigenvalue \({\lambda _i}.\) In practice, \(k\) can be taken as the algebraic multiplicity of \({\lambda _i}.\)

Similar rules for determining the degree of the polynomials are used for quasi-polynomials of kind

\[{e^{\alpha t}}\cos \left( {\beta t} \right),\;\; {e^{\alpha t}}\sin\left( {\beta t} \right).\]

Here the resonance case occurs when the number \(\alpha + \beta i\) coincides with a complex eigenvalue \({\lambda _i}\) of the matrix \(A.\)

After the structure of a particular solution \({\mathbf{X}_1}\left( t \right)\) is chosen, the unknown vector coefficients \({A_0},\) \({A_1}, \ldots ,\) \({A_m}, \ldots ,\) \({A_{m + k}}\) are found by substituting the expression for \({\mathbf{X}_1}\left( t \right)\) in the original system and equating the coefficients of the terms with equal powers of \(t\) on the left and right side of each equation.

Method of Variation of Constants

The method of variation of constants (Lagrange method) is the common method of solution in the case of an arbitrary right-hand side \(\mathbf{f}\left( t \right).\)

Suppose that the general solution of the associated homogeneous system is found and represented as

\[{\mathbf{X}_0}\left( t \right) = \Phi \left( t \right)\mathbf{C},\]

where \(\Phi \left( t \right)\) is a fundamental system of solutions, i.e. a matrix of size \(n \times n,\) whose columns are formed by linearly independent solutions of the homogeneous system, and \(\mathbf{C} = {\left( {{C_1},{C_2}, \ldots ,{C_n}} \right)^T}\) is the vector of arbitrary constant numbers \({C_i}\left( {i = 1, \ldots ,n} \right).\)

We replace the constants \({C_i}\) with unknown functions \({C_i}\left( t \right)\) and substitute the function \(\mathbf{X}\left( t \right) = \Phi \left( t \right)\mathbf{C}\left( t \right)\) in the nonhomogeneous system of equations:

\[\mathbf{X'}\left( t \right) = A\mathbf{X}\left( t \right) + \mathbf{f}\left( t \right),\;\; \Rightarrow
\cancel{\Phi'\left( t \right)\mathbf{C}\left( t \right)} + \Phi \left( t \right)\mathbf{C'}\left( t \right) = \cancel{A\Phi \left( t \right)\mathbf{C}\left( t \right)} + \mathbf{f}\left( t \right),\;\; \Rightarrow \Phi \left( t \right)\mathbf{C'}\left( t \right) = \mathbf{f}\left( t \right).\]

Since the Wronskian of the system is not equal to zero, then there exists the inverse matrix \({\Phi ^{ - 1}}\left( t \right).\) Multiplying the last equation on the left by \({\Phi ^{ - 1}}\left( t \right),\) we obtain:

\[{\Phi ^{ - 1}}\left( t \right)\Phi \left( t \right)\mathbf{C'}\left( t \right) = {\Phi ^{ - 1}}\left( t \right)\mathbf{f}\left( t \right),\;\; \Rightarrow \mathbf{C'}\left( t \right) = {\Phi ^{ - 1}}\left( t \right)\mathbf{f}\left( t \right),\;\; \Rightarrow \mathbf{C}\left( t \right) = {\mathbf{C}_0} + \int {{\Phi ^{ - 1}}\left( t \right)\mathbf{f}\left( t \right)dt} ,\]

where \({\mathbf{C}_0}\) is an arbitrary constant vector.

Then the general solution of the nonhomogeneous system can be written as

\[\mathbf{X}\left( t \right) = \Phi \left( t \right)\mathbf{C}\left( t \right) = \Phi \left( t \right){\mathbf{C}_0} + \Phi \left( t \right)\int {{\Phi ^{ - 1}}\left( t \right)\mathbf{f}\left( t \right)dt} = {\mathbf{X}_0}\left( t \right) + {\mathbf{X}_1}\left( t \right).\]

We see that a particular solution of the nonhomogeneous equation is represented by the formula

\[{\mathbf{X}_1}\left( t \right) = \Phi \left( t \right)\int {{\Phi ^{ - 1}}\left( t \right)\mathbf{f}\left( t \right)dt}.\]

Thus, the solution of the nonhomogeneous equation can be expressed in quadratures for any inhomogeneous term \(\mathbf{f}\left( t \right).\) In many problems, the corresponding integrals can be calculated analytically. This allows us to express the solution of the nonhomogeneous system explicitly.